THE AUTOMORPHISM TOWER PROBLEM REVISITED

WINFRIED JUST, SAHARON SHELAH, AND SIMON THOMAS

Abstract. It is well-known that the automorphism towers of infinite centreless groups of cardinality κ terminate in less than (2^{κ})^{+} steps. But an easy counting argument shows that (2^{κ})^{+} is not the best possible bound. However, in this paper, we will show that it is impossible to find an explicit better bound using ZF C.

1. Introduction

If G is a centreless group, then there is a natural embedding of G into its automorphism group Aut G, obtained by sending each g ∈ G to the corresponding inner automorphism i_{g} ∈ Aut G. In this paper, we will always work with the left action of Aut G on G. Thus i_{g}(x) = gxg^{−1} for all x ∈ G. If π ∈ Aut G and g ∈ G, then an easy calculation shows that πi_{g}π^{−1} = i_{π(g)}. Hence the group of inner automorphisms Inn G is a normal subgroup of Aut G; and C_{Aut G}(Inn G) = 1. In particular, Aut G is also a centreless group. This enables us to define the automorphism tower of G to be the ascending chain of groups

G = G_{0} ⊴ G_{1} ⊴ G_{2} ⊴ … G_{α} ⊴ G_{α+1} ⊴ …

such that for each ordinal α

(a) G_{α+1} = Aut G_{α}; and

(b) if α is a limit ordinal, then G_{α} = ⋃_{β<α}G_{β}. (At each successor step, we identify G_{α} with Inn G_{α} via the natural embedding.)

The automorphism tower is said to terminate if there exists an ordinal α such that G_{α+1} = G_{α}. This occurs if and only if there exists an ordinal α such that Aut G_{α} = Inn G_{α}. A classical result of Wielandt [15] says that if G is finite, then

The research of the first author was done during a visit to Rutgers University in September 1997, which was partially supported by NSF Grant DMS-9704477.

The research of the second author was partially supported by the U.S.-Israel Binational Science Foundation. This paper is number 654 in the cumulative list of the second author’s publications.

The research of the third author was partially supported by NSF Grants.

2 WINFRIED JUST, SAHARON SHELAH, AND SIMON THOMAS

the automorphism tower terminates after finitely many steps. Wielandt’s theorem fails for infinite centreless groups. For example, consider the infinite dihedral group D_{∞} = ⟨a, b⟩, where a and b are elements of order 2. Then D_{∞} = ⟨a⟩ ∗ ⟨b⟩ is the free product of its cyclic subgroups ⟨a⟩ and ⟨b⟩. It follows that D_{∞} is a centreless group; and that D_{∞} has an outer automorphism π of order 2 which interchanges the elements a and b. It is easily shown that Aut D_{∞} = ⟨π, i_{a}⟩. Thus Aut D_{∞} is also an infinite dihedral group, and so Aut D_{∞} ≃ D_{∞}. Hence for each n ∈ ω, the n^{th} group in the automorphism tower of D_{∞} is isomorphic to D_{∞}; and the automorphism tower of D_{∞} does not terminate after finitely many steps.

In the 1970s, a number of special cases of the automorphism tower problem were solved. For example, Rae and Roseblade [8] proved that the automorphism tower of a centreless Černikov group terminates after finitely many steps; and Hulse [4] proved that the automorphism tower of a centreless polycyclic group terminates in countably many steps. But the problem was not solved in full generality until 1984, when Thomas [13] showed that the automorphism tower of an arbitrary centreless group eventually terminates; and that for each ordinal α, there exists a group whose automorphism tower terminates in exactly α steps.

Definition 1.1. If G is a centreless group, then the height τ(G) of the automorphism tower of G is the least ordinal α such that G_{α+1} = G_{α}.

This raises the question of finding bounds for τ(G) in terms of the cardinality of G. In his original paper [13], Thomas proved that if G is an infinite centreless group of cardinality κ, then τ(G) ≤ (2^{κ})^{+}. Soon afterwards, Thomas and Felgner independently noticed that an easy application of Fodor’s Lemma yielded the following slightly better bound.

Theorem 1.2 (Thomas [14]). If G is an infinite centreless group of cardinality κ, then τ(G) < (2^{κ})^{+}.

□

Definition 1.3. If κ is an infinite cardinal, then τ_{κ} is the least ordinal such that τ(G) < τ_{κ} for every centreless group G of cardinality κ.

Since there are only 2^{κ} centreless groups of cardinality κ up to isomorphism, it follows that τ_{κ} < (2^{κ})^{+}. On the other hand, Thomas [13] has shown that for

THE AUTOMORPHISM TOWER PROBLEM REVISITED 3

each ordinal α < κ^{+}, there exists a centreless group G of cardinality κ such that τ(G) = α. Thus κ^{+} ≤ τ_{κ} < (2^{κ})^{+}. It is natural to ask whether a better explicit bound on τ_{κ} can be proved in ZF C, preferably one which does not involve cardinal exponentiation.

The proof of Theorem 1.2 is extremely simple, and uses only the most basic results in group theory, together with some elementary properties of the infinite cardinal numbers. So it is not surprising that Theorem 1.2 does not give the best possible bound for τ_{κ}. In contrast, the proof of Wielandt’s theorem is much deeper, and involves an intricate study of the subnormal subgroups of a finite centreless group. The real question behind the search for better explicit bounds for τ_{κ} is whether there exists a subtler, more informative, group-theoretic proof of the automorphism tower theorem for infinite groups. The main result of this paper says that no such bounds can be proved in ZF C, and thus can be interpreted as saying that no such proof exists. (It is perhaps worth mentioning that the proof of Theorem 1.2 yields that the automorphism tower of a finite centreless group terminates in countably many steps. However, there does not seem to be an easy reduction from countable to finite; and it appears that some form of Wielandt’s analysis is necessary.)

Theorem 1.4. Let V ⊨ GCH and let κ, λ ∈ V be uncountable cardinals such that κ < cf(λ). Let α be any ordinal such that α < λ^{+}. Then there exists a notion of forcing ℙ, which preserves cofinalities and cardinalities, such that the following statements are true in the corresponding generic extension V^{ℙ}.

(a) 2^{κ} = λ.

(b) There exists a centreless group G of cardinality κ such that τ(G) = α.

Thus it is impossible to find better explicit bounds for τ_{κ} when κ is an uncountable cardinal. However, our methods do not enable us to deal with countable groups; and it remains an open question whether or not there exists a countable centreless group G such that τ(G) ≥ ω_{1}.

Most of this paper will be concerned with the problem of constructing centreless groups with extremely long automorphism towers. Unfortunately it is usually very difficult to compute the successive groups in an automorphism tower. We will get

4 WINFRIED JUST, SAHARON SHELAH, AND SIMON THOMAS

around this difficulty by reducing it to the much easier computation of the successive normalisers of a subgroup H of a group G.

Definition 1.5. If H is a subgroup of the group G, then the normaliser tower of H in G is defined inductively as follows.

(a) N_{0}(H) = H.

(b) If α = β + 1, then N_{α}(H) = N_{G} (N_{β}(H)).

(c) If α is a limit ordinal, then N_{α}(H) = ⋃_{β<α}N_{β}(H). It is sometimes necessary for the notation to include an explicit reference to the ambient group G. In this case, we will write N_{α}(H) = N_{α}(H, G).

As we will see in Section 2, if α is any ordinal, then it is easy to construct examples of pairs of groups, H < G, such that the normaliser tower of H in G terminates in exactly α steps. The following lemma, which was essentially proved in [13], will enable us to convert normaliser towers into corresponding automorphism towers.

Lemma 1.6. Let K be a field such that |K| > 3 and let H be a subgroup of Aut K. Let

G = P GL(2, K) ⋊ H ⩽ P Γ L(2, K) = P GL(2, K) ⋊ Aut K.

Then G is a centreless group; and for each α, G_{α} = P GL(2, K) ⋊ N_{α}(H), where N_{α}(H) is the α^{th} group in the normaliser tower of H in Aut K.

□

It is well-known that every group G can be realised as the automorphism group of a suitable graph Γ. Thus the following result implies that every group G can also be realised as the automorphism group of a suitable field K.

Lemma 1.7 (Fried and Kollár [2]). Let Γ = ⟨X, E⟩ be any graph. Then there exists a field K_{Γ} of cardinality max{|X| , ω} which satisfies the following conditions.

(a) X is an Aut K_{Γ}-invariant subset of K_{Γ}.

(b) The restriction mapping, π ↦ π ↾ X, is an isomorphism from Aut K_{Γ} onto Aut Γ.

□

THE AUTOMORPHISM TOWER PROBLEM REVISITED 5

Combining Lemmas 1.6 and 1.7, we obtain the following reduction of our problem.

Lemma 1.8. Suppose that there exists a graph Γ of cardinality κ and a subgroup H of Aut Γ such that

(a) |H| ≤ κ; and

(b) the normaliser tower of H in Aut Γ terminates in exactly α steps. Then there exists a centreless group T of cardinality κ such that τ(T) = α.

Proof. Let K_{Γ} be the corresponding field, which is given by Lemma 1.7, and let T = P GL(2, K_{Γ}) ⋊ H. By Lemma 1.6, τ(T) = α.

From now on, fix a regular uncountable cardinal κ such that κ^{<κ} = κ and an ordinal α. Roughly speaking, our strategy will be to

(1) first construct a pair of groups, H < G, such that |H| ≤ κ and the normaliser tower of H in G terminates in α steps; and

(2) then attempt to find a cardinal-preserving notion of forcing ℙ which adjoins a graph Γ of cardinality κ such that G ≃ Aut Γ.

Of course, there are many groups G for which such a notion of forcing ℙ cannot possibly exist. For example, De Bruijn [1] has shown that the alternating group Alt(κ^{+}) cannot be embedded in Sym(κ). Consequently, there is no cardinalpreserving notion of forcing ℙ which adjoins a graph Γ of cardinality κ such that Alt(κ^{+}) ≃ Aut Γ.

Our next definition singles out a combinatorial condition which is satisfied by all those groups G such that G is embeddable in Sym(κ). (See Proposition 1.11.) Conversely, in Theorem 1.12, we will show that if a group G satisfies this combinatorial condition, then there exists a cardinal-preserving notion of forcing ℙ which adjoins a graph Γ of cardinality κ such that G ≃ Aut Γ.

Definition 1.9. Let κ be a regular uncountable cardinal such that κ^{<κ} = κ. Then a group G is said to satisfy the κ^{+}-compatibility condition if it has the following property. Suppose that H is a group such that |H| < κ. Suppose that (f_{i} | i < κ^{+}) is a sequence of embeddings f_{i} : H → G; and let H_{i} = f_{i}[H] for each i < κ^{+}. Then there exist ordinals i < j < κ^{+} and a surjective homomorphism φ : ⟨H_{i}, H_{j}⟩ → H_{i} such that (a) φ ∘ f_{j} = f_{i}; and

(b) φ ↾ H_{i} = id_{H_{i}}.

Example 1.10. To get an understanding of Definition 1.9, it will probably be helpful to see an example of a group which fails to satisfy the κ^{+}-compatibility condition. So we will show that Alt(κ^{+}) does not satisfy the κ^{+}-compatibility condition. Let H = Alt(4). For each 3 ≤ i < κ^{+}, let Δ_{i} = {0, 1, 2, i} and let f_{i} : Alt(4) → Alt(Δ_{i}) be an isomorphism. If 3 ≤ i < j < κ^{+}, then

⟨Alt(Δ_{i}), Alt(Δ_{j})⟩ = Alt(Δ_{i} ∪ Δ_{j}) ≃ Alt(5).

Since Alt(5) is a simple group, there does not exist a surjective homomorphism from ⟨Alt(Δ_{i}), Alt(Δ_{j})⟩ onto Alt(Δ_{i}).

Proposition 1.11. Let κ be a regular uncountable cardinal such that κ^{<κ} = κ, and let G ⩽ Sym(κ). Then G satisfies the κ^{+}-compatibility condition.

Proof. Let H be a group such that |H| < κ, and let (f_{i} | i < κ^{+}) be a sequence of embeddings f_{i} : H → G. For each i < κ^{+}, let H_{i} = f_{i}[H]; and let Z_{i} be a subset of κ chosen so that

(a) |Z_{i}| < κ;

(b) Z_{i} is H_{i}-invariant; and

(c) g ↾ Z_{i} ≠ id_{Z_{i}} for all 1 ≠ g ∈ H_{i}. After passing to a suitable subsequence if necessary, we can suppose that the following conditions hold.

(1) There exists a fixed subset Z such that Z_{i} = Z for all i < κ^{+}.

(2) For each i < κ^{+}, let r_{i} : H_{i} → Sym(Z) be the restriction mapping, g ↦ g ↾ Z; and let π_{i} : H → Sym(Z) be the embedding defined by π_{i} = r_{i} ∘ f_{i}. Then π_{i} = π_{j} for all i < j < κ^{+}.

Fix any pair of ordinals i, j such that i < j < κ^{+}. Let ρ : ⟨H_{i}, H_{j}⟩ → Sym(Z) be the restriction mapping, g ↦ g ↾ Z. Then ρ [⟨H_{i}, H_{j}⟩] = r_{i} [H_{i}], and so we can define a surjective homomorphism φ : ⟨H_{i}, H_{j}⟩ → H_{i} by φ = r_{i}^{−1} ∘ ρ. Clearly φ ↾ H_{i} = id_{H_{i}}; and it is easily checked that φ ∘ f_{j} = f_{i}.

THE AUTOMORPHISM TOWER PROBLEM REVISITED 7

Theorem 1.12. Let κ be a regular uncountable cardinal such that κ^{<κ} = κ, and let G be a group which satisfies the κ^{+}-compatibility condition. Then there exists a notion of forcing ℙ such that

(a) ℙ is κ-closed;

(b) ℙ has the κ^{+}- c.c.; and

(c) ⊩_{ℙ} There exists a graph Γ of cardinality κ such that G ≃ Aut Γ. Furthermore, if |G| = λ, then |ℙ| = max{κ, λ^{<κ}}.

Combining Proposition 1.11 and Theorem 1.12, we see that if κ is an uncountable cardinal such that κ^{<κ} = κ and G is an arbitrary subgroup of Sym(κ), then there exists a cardinal-preserving notion of forcing ℙ and a graph Γ ∈ V^{ℙ} such that G ≃ Aut Γ. This result is not true of arbitrary subgroups of Sym(ω); for Solecki [11] has shown that no uncountable free abelian group is the automorphism group of a countable first-order structure.

Theorem 1.12 will be proved in Section 3. It is now easy to explain the main points of the proof of Theorem 1.4. Assume that V ⊨ GCH. Let κ be a regular uncountable cardinal, and let λ be a cardinal such that cf(λ) > κ. Let α be any ordinal such that α < λ^{+}. In Section 2, we will prove that there exists a notion of forcing ℚ such that

(1) ℚ is κ-closed;

(2) ℚ has the κ^{+}- c.c.; and such that the following statements are true in the generic extension V^{ℚ}.

(a) 2^{κ} = λ.

(b) There exist groups H < G < Sym(κ) such that |H| = κ and the normaliser tower of H in G terminates in exactly α steps.

By Proposition 1.11, G satisfies the κ^{+}-compatibility condition. Hence we can use Theorem 1.12 to generically adjoin a graph Γ of cardinality κ such that G ≃ Aut Γ. A moment’s thought shows that the normaliser tower of H in G is an absolute notion. Thus Lemma 1.8 yields a centreless group T of cardinality κ such that τ(T) = α. The case when κ is a singular cardinal requires a little more work, and will be dealt with in Section 4. The remainder of this section will be devoted to another two easy applications of Theorem 1.12.

8 WINFRIED JUST, SAHARON SHELAH, AND SIMON THOMAS

Application 1.13. A well-known open problem asks whether there exists a countable structure M such that Aut M is the free group on 2^{ω} generators. Using Theorem 1.12, it is easy to establish the consistency of the existence of a structure N of cardinality ω_{1} such that Aut N is the free group on 2^{ω_{1}} generators. It is not known whether the existence of such a structure can be proved in ZF C.

Theorem 1.14. Let V be a transitive model of ZF C and let κ, λ, θ be cardinals such that κ^{<κ} = κ < λ ≤ θ = θ^{κ}. Then there exists a notion of forcing ℙ, which preserves cofinalities and cardinalities, such that the following statements are true in V^{ℙ}.

(a) 2^{κ} = θ; and

(b) there exists a graph Γ of cardinality κ such that Aut Γ is the free group on λ generators.

Proof. After performing a preliminary forcing if necessary, we can also suppose that 2^{κ} = θ. Let F be the free group on λ generators. Then it is enough to show that F satisfies the κ^{+}-compatibility condition. Let H be a (necessarily free) group such that |H| < κ, and let (f_{i} | i < κ^{+}) be a sequence of embeddings f_{i} : H → F. For each i < κ^{+}, let H_{i} = f_{i}[H]. Then ⟨H_{i} | i < κ^{+}⟩ is a free group of cardinality at most κ^{+}. By Theorem 5.1 [10], there exists an embedding of ⟨H_{i} | i < κ^{+}⟩ into Sym(κ). So Proposition 1.11 yields the existence of ordinals i < j < κ^{+} and a surjective homomorphism φ : ⟨H_{i}, H_{j}⟩ → H_{i} such that φ ∘ f_{j} = f_{i} and φ ↾ H_{i} = id_{H_{i}}.

Application 1.15. Theorem 1.6 [14] says that if G is a finitely generated centreless group, then the automorphism tower of G terminates in countably many steps. It is conceivable that a more general result holds; namely, that the automorphism tower of G terminates in countably many steps, whenever G is a countable centreless group such that Aut G is also countable. To see why this might be true, let G be such a group. Then, by Kueker [5], there exists a finite subset F ⊆ G such that each automorphism π ∈ Aut G is uniquely determined by its restriction π ↾ F. In terms of the automorphism tower of G, this says that there is a finite subset F ⊆ G such that C_{G_{1}}(F) = 1. Suppose that the “rigidity” of F within G = G_{0} is propagated along the automorphism tower of G; ie. that C_{G_{α}}(F) = 1 for all ordinals α < ω_{1}. Then the proof of Theorem 1.6 [14] shows that the automorphism tower of G terminates in countably many steps.

Question 1.16. Let G be a centreless group such that |Aut G| = ω. Does there exist a finite subset F ⊆ G such that C_{G_{α}}(F) = 1 for all ordinals α < ω_{1}?

If |G_{α}| = ω for all α < ω_{1}, then Fodor’s Lemma implies that there exists an ordinal β < ω_{1} and a finite subset F_{β} of G_{β} such that C_{G_{α}}(F_{β}) = 1 for all β ≤ α < ω_{1}; and so τ(G) < ω_{1}. This observation suggests the following weak form of Question 1.16, which is also open.

Question 1.17. Does there exist a centreless group G such that |Aut G| = ω and |Aut(Aut G)| = 2^{ω}?

Of course, a positive answer to Question 1.16 implies a negative answer to Question 1.17. Using Theorem 1.12, it is easy to establish the consistency of the existence of a centreless group G of cardinality ω_{1} such that |Aut G| = ω_{1} and |Aut(Aut G)| = 2^{ω_{1}}. Once again, it is not known whether the existence of such a group can be proved in ZF C.

Theorem 1.18. Let κ be a regular uncountable cardinal such that κ^{<κ} = κ. Then it is consistent that there exists a centreless group G of cardinality κ such that |Aut G| = κ and |Aut(Aut G)| = 2^{κ}.

Proof. Let V be the ground model. For each α, ξ < κ, let Z_{ξ}^{α} = ⟨z_{ξ}^{α}⟩ be an infinite cyclic group. For each ξ < κ, let A_{ξ} = ⨁_{α<κ}Z_{ξ}^{α}; and let B = ⨁_{ξ<κ}A_{ξ}. Define an action of Sym(κ) on B by πz_{ξ}^{α}π^{−1} = z_{π(ξ)}^{α} for all α, ξ < κ; and let W = B ⋊ Sym(κ) be the corresponding semidirect product. Let H = ⨁_{ξ<κ}Z_{ξ}^{ξ}. Then the members of the normaliser tower of H in W are

(a) N_{0}(H) = H;

(b) N_{1}(H) = B;

(c) N_{2}(H) = W. Clearly W is embeddable in Sym(κ); and so W satisfies the κ^{+}-compatibility condition. Let ℙ be the notion of forcing, given by Theorem 1.12, which adjoins a graph Γ of cardinality κ such that W ≃ Aut Γ. Let K_{Γ} ∈ V^{ℙ} be the corresponding field,

10 WINFRIED JUST, SAHARON SHELAH, AND SIMON THOMAS

which is given by Lemma 1.7. Then G = P GL(2, K_{Γ}) ⋊ H is a group such that

|Aut G| = |P GL(2, K_{Γ}) ⋊ B| = κ

and

|Aut(Aut G)| = |P GL(2, K_{Γ}) ⋊ W | = 2^{κ}.

Our set-theoretic notation mainly follows that of Kunen [6]. Thus if ℙ is a notion of forcing and p,q ∈ ℙ, then q ≤ p means that q is a strengthening of p. We say that ℙ is κ-closed if for every λ < κ, every descending sequence of elements of ℙ

p_{0} ≥ p_{1} ≥ ⋯ ≥ p_{ξ} ≥ …, ξ < λ,

has a lower bound in ℙ. If V is the ground model, then we will denote the generic extension by V^{ℙ} if we do not wish to specify a particular generic filter H ⊆ ℙ. If we want to emphasize that the term t is to be interpreted in the model M of ZF C, then we write t^{M}; for example, Sym(λ)^{M}.

Our group-theoretic notation is standard. For example, the (restricted) wreath product of A by C is denoted by AwrC; and the direct sum of the groups H_{ξ}, ξ < λ, is denoted by ⨁_{ξ<λ}H_{ξ}. If π ∈ Sym(κ), then supp(π) = {α ∈ κ | π(α) ≠ α}; and if λ is an infinite cardinal such that λ ≤ κ, then Sym_{λ}(κ) = {π ∈ Sym(κ) | |supp(π)| < λ}.

2. Realising normaliser towers within infinite symmetric groups

Let κ be a regular uncountable cardinal such that κ^{<κ} = κ. In this section, we will study the problem of realising long normaliser towers within Sym(κ). In particular, we will prove that if α is any ordinal, then there exists a generic extension V^{ℚ} such that a normaliser tower of height α can be realised in Sym(κ)^{V^{ℚ}}.

First for each ordinal α, we will construct a pair of groups, H < G, such that the normaliser tower of H in G terminates in exactly α steps.

Definition 2.1. The ascending chain of groups

W_{0} ⩽ W_{1} ⩽ ⋯ ⩽ W_{α} ⩽ W_{α+1} ⩽ …

is defined inductively as follows.

(a) W_{0} = C_{2}, the cyclic group of order 2.

(b) Suppose that α = β + 1. Then

W_{β} = W_{β} ⊕ 1 ⩽ [W_{β} ⊕ W_{β}^{∗}] ⋊ ⟨σ_{β+1}⟩ = W_{β+1}.

Here W_{β}^{∗} is an isomorphic copy of W_{β}; and σ_{β+1} is an element of order 2 which interchanges the factors W_{β} ⊕1 and 1⊕W_{β}^{∗} of the direct sum W_{β} ⊕W_{β}^{∗} via conjugation. Thus W_{β+1} is isomorphic to the wreath product W_{β}wrC_{2}.

(c) If α is a limit ordinal, then W_{α} = ⋃_{β<α}W_{β}.

Lemma 2.2. |W_{α}| ≤ max{|α|, ω} for all ordinals α.

Proof. This follows by an easy induction on α.

Lemma 2.3. (a) If 1 ≤ n < ω, then the normaliser tower of W_{0} in W_{n} termi-

nates in exactly n + 1 steps.

(b) If α ≥ ω, then the normaliser tower of W_{0} in W_{α} terminates in exactly α steps.

Proof. (a) It is easily checked that

N_{1}(W_{0}, W_{n}) = W_{0} ⊕ W_{0}^{∗} ⊕ W_{1}^{∗} ⊕ ⋯ ⊕ W_{n−1}^{∗}

and that

N_{2}(W_{0}, W_{n}) = W_{1} ⊕ W_{1}^{∗} ⊕ ⋯ ⊕ W_{n−1}^{∗};

and that, in general, for each 0 ≤ ℓ ≤ n − 1,

N_{ℓ+1}(W_{0}, W_{n}) = W_{ℓ} ⊕ ⨁_{ℓ≤m<n}W_{m}^{∗}.

(b) For example, consider the case when α > ω. Then for each ℓ ∈ ω,

N_{ℓ+1}(W_{0}, W_{α}) = W_{ℓ} ⊕ ⨁_{ℓ≤β<α}W_{β}^{∗};

and for each γ such that ω ≤ γ < α,

N_{γ}(W_{0}, W_{α}) = W_{γ} ⊕ ⨁_{γ≤β<α}W_{β}^{∗}.

12 WINFRIED JUST, SAHARON SHELAH, AND SIMON THOMAS

Remark 2.4. Unfortunately, the group W_{κ^{+}} does not satisfy the κ^{+}-compatibility condition. To see this, let

H = C_{2}wrC_{2} = [⟨a⟩ ⊕ ⟨b⟩] ⋊ ⟨c⟩;

and for each limit ordinal i < κ^{+}, let f_{i} : H → W_{κ^{+}} be the embedding such that f_{i}(a) = σ_{i+1} and f_{i}(c) = σ_{i+2}. (Here we are using the notation which was introduced in Definition 2.1.) Let i, j be any limit ordinals such that i < j < κ^{+}. Then

⟨H_{i}, H_{j}⟩ ≃ ((C_{2}wrC_{2}) wrC_{2}) wrC_{2}.

Suppose that there exists a a surjective homomorphism φ : ⟨H_{i}, H_{j}⟩ → H_{i} such that

(a) φ ∘ f_{j} = f_{i}; and

(b) φ ↾ H_{i} = id_{H_{i}}. Then φ(σ_{j+1}) = φ(σ_{i+1}) = σ_{i+1} and φ(σ_{j+2}) = φ(σ_{i+2}) = σ_{i+2}. Consider the element x = σ_{j+1}σ_{j+2}σ_{j+1}σ_{j+2} ∈ H_{j}. Then it is easily checked that

(1) x lies in the centre of H_{j}; and

(2) σ_{j+1}yσ_{j+1}^{−1} = xyx^{−1} for all y ∈ H_{i}. Thus z = φ(x) lies in the centre of H_{i}. Since φ(σ_{j+1}) = σ_{i+1}, we find that

σ_{i+1}yσ_{i+1}^{−1} = zyz^{−1} = y

for all y ∈ H_{i}. But this contradicts the fact that σ_{i+1} is a noncentral element of H_{i}.

Thus if α ≥ κ^{+}, then W_{α} is not embeddable in Sym(κ). However, the above argument does not rule out the possibility that W_{α} is embeddable in the quotient group Sym(κ)/ Sym_{κ}(κ); and this is enough for our purposes.

Lemma 2.5. Let κ be an infinite cardinal such that κ^{<κ} = κ. Suppose that γ is an ordinal, and that there exists an embedding

f : W_{γ} → Sym(κ)/ Sym_{κ}(κ).

Then for each ordinal α ≤ γ, there exist groups H ⩽ G ⩽ Sym(κ) such that |H| = κ and the normaliser tower of H in G terminates in exactly α steps.

Proof. For each α ≤ γ, let H^{α} be the subgroup of Sym(κ) such that f [W_{α}] = H^{α}/ Sym_{κ}(κ). Since | Sym_{κ}(κ)| = κ^{<κ} = κ, it follows that |H^{0}| = κ.

Claim 2.6. For each ordinal β,

f [N_{β}(W_{0}, W_{α})] = N_{β}(H^{0}, H^{α})/ Sym_{κ}(κ).

Proof. We will argue by induction on β. The result is clear when β = 0, and no difficulties arise when β is a limit ordinal. Suppose that β = ξ + 1 and that the result holds for ξ. Let R = N_{ξ}(H^{0}, H^{α}); and for each subgroup K such that Sym_{κ}(κ) ⩽ K ⩽ H^{α}, let \overline{K} = K/ Sym_{κ}(κ). Then

f [N_{ξ+1}(W_{0}, W_{α})] = N\overline{H^{α}}(\overline{R});

and so we must show that

N\overline{H^{α}}(\overline{R}) = \overline{N_{H^{α}}(R)}.

But this is an immediate consequence of the Correspondence Theorem for subgroups of quotient groups, together with the observation that the normaliser of any subgroup L is the largest subgroup M such that L ⊴ M.

It is now easy to complete the proof of Lemma 2.5. Applying Lemma 2.3 and Claim 2.6, we see that if α ≥ ω, then the normaliser tower of H^{0} in H^{α} terminates in exactly α steps; and that if 2 ≤ α = n < ω, then the normaliser tower of H^{0} in H^{n−1} terminates in exactly n steps. This just leaves the cases when α = 0, 1. When α = 0, we can take H = G = Alt(κ); and when α = 1, we can take H = Alt(κ) and G = Sym(κ).

The next result implies that if ω < κ^{κ} = κ and W is any group, then there exists a cardinal-preserving notion of forcing ℙ such that in V^{ℙ}, the group W is embeddable in Sym(κ)/ Sym_{κ}(κ). (If |W |^{κ} > |W |, then just embed W in a larger group L such that |L|^{κ} = |L|; and then apply Lemma 2.7 to L.)

Lemma 2.7. Let V be a transitive model of ZF C and let κ, λ be cardinals such that ω < κ^{<κ} = κ < λ = λ^{κ}. Let W be any group of cardinality λ. Then there exists a notion of forcing ℚ such that

(1) ℚ is κ-closed.

(2) ℚ has the κ^{+}- c.c.;

14 WINFRIED JUST, SAHARON SHELAH, AND SIMON THOMAS

and such that the following statements are true in the generic extension V^{ℚ}.

(a) 2^{κ} = λ.

(b) There exists an isomorphic embedding

f : W → (Sym(κ)/ Sym_{κ}(κ))^{V^{ℚ}} .

Proof. Let Ω = ⋃_{α<κ}{α} × α. We will work with the symmetric group Sym(Ω) rather than with Sym(κ). Let ℚ be the notion of forcing consisting of the conditions

p = (δ_{p}, H_{p}, E_{p})

such that the following hold.

(a) ω ≤ δ_{p} < κ.

(b) H_{p} is a subgroup of W such that |H_{p}| ≤ |δ_{p}|.

(c) E_{p} is a function which assigns a permutation e_{π,ξ}^{p} ∈ Sym({ξ}×ξ) to each pair (π, ξ) ∈ H_{p} × δ_{p}.

We set q = (δ_{q}, H_{q}, E_{q}) ≤ p = (δ_{p}, H_{p}, E_{p}) if and only if

(1) δ_{p} ≤ δ_{q};

(2) H_{p} ⩽ H_{q};

(3) E_{p} ⊆ E_{q}; and

(4) if δ_{p} ≤ ξ < δ_{q}, then the restriction to H_{p} of the function, π ↦ e_{π,ξ}^{q}, is an isomorphic embedding of H_{p} into Sym({ξ} × ξ).

Claim 2.8. ℚ is κ-closed.

Proof of Claim 2.8. This is clear.

Claim 2.9. ℚ has the κ^{+}- c.c..

Proof of Claim 2.9. Suppose that p_{i} = (δ_{p_{i}}, H_{p_{i}}, E_{p_{i}}) ∈ ℚ for i < κ^{+}. Using the Δ-System Lemma and the assumption that κ^{<κ} = κ, after passing to a suitable subsequence if necessary, we can suppose that the following conditions are satisfied.

(i) There exists a fixed ordinal δ such that δ_{p_{i}} = δ for all i < κ^{+}. (ii) There exists a fixed subgroup H such that H_{p_{i}} ∩ H_{p_{j}} = H for all i < j < κ^{+}. (iii) There exists a fixed function E such that E_{p_{i}} ↾ H = E for all i < κ^{+}.

THE AUTOMORPHISM TOWER PROBLEM REVISITED 15

Now fix any two ordinals i < j < κ^{+}. Let H^{+} = ⟨H_{p_{i}}, H_{p_{j}}⟩ be the subgroup generated by H_{p_{i}} ∪ H_{p_{j}}; and let E^{+} be any extension of E_{p_{i}} ∪ E_{p_{j}} which satisfies condition (c). Then q = (δ, H^{+}, E^{+}) is a common lower bound of p_{i} and p_{j}.

Claim 2.10. For each α < κ, the set C_{α} = {p ∈ ℚ | δ_{p} ≥ α} is dense in ℚ.

Proof of Claim 2.10. Let p = (δ_{p}, H_{p}, E_{p}) ∈ ℚ. Then we can suppose that δ_{p} < α. We can define an isomorphic embedding φ : H_{p} → Sym(H_{p}) by setting φ(h)(x) = hx for all x ∈ H_{p}. Since |H_{p}| ≤ |δ_{p}|, it follows that there exists an isomorphic embedding φ_{ξ} : H_{p} → Sym({ξ} × ξ) for each δ_{p} ≤ ξ < α. Hence there exists a condition q = (δ_{q}, H_{q}, E_{q}) ≤ p such that H_{q} = H_{p} and δ_{q} = α.

Claim 2.11. For each π ∈ W, the set D_{π} = {p ∈ ℚ | π ∈ H_{p}} is dense in ℚ.

Proof of Claim 2.11. Let p = (δ_{p}, H_{p}, E_{p}) ∈ ℚ. Let H^{+} = ⟨H_{p}, π⟩ be the subgroup generated by H_{p} ∪ {π}, and let E^{+} be any extension of E_{p} to H^{+} which satisfies condition (c). Then q = (δ_{q}, H_{q}, E_{q}) ≤ p.

Let F ⊆ ℚ be a generic filter, and let V^{ℚ} = V [F] be the corresponding generic extension. Working within V^{ℚ}, for each π ∈ W, let

e(π) = ⋃{e_{π,ξ}^{p} | There exists p ∈ F such that π ∈ H_{p} and ξ < δ_{p}}.

Then e(π) ∈ Sym(Ω). Let Sym_{κ}(Ω) = {ψ ∈ Sym(Ω) | | supp(ψ)| < κ}; and define the function

f : W → Sym(Ω)/ Sym_{κ}(Ω)

by f(π) = e(π) Sym_{κ}(Ω). Then it is enough to show that f is an isomorphic embedding.

Claim 2.12. If 1 ≠ π ∈ W, then f(π) ≠ 1.

Proof of Claim 2.12. Choose a condition p = (δ_{p}, H_{p}, E_{p}) ∈ F such that π ∈ H_{p}. If ξ is any ordinal such that δ_{p} ≤ ξ < κ, then e(π) ↾ {ξ} × ξ ≠ id_{{ξ}×ξ}. Hence e(π) ∉ Sym_{κ}(Ω).

Claim 2.13. f is a group homomorphism.

16 WINFRIED JUST, SAHARON SHELAH, AND SIMON THOMAS

Proof of Claim 2.13. Let π_{1}, π_{2} ∈ W. Let p = (δ_{p}, H_{p}, E_{p}) ∈ F be a condition such that π_{1}, π_{2} ∈ H_{p}. Let ξ be any ordinal such that δ_{p} ≤ ξ < κ; and let q ∈ F be a condition such that q ≤ p and ξ < δ_{q}. Then

e_{π_{1},ξ}^{q} ∘ e_{π_{2},ξ}^{q} = e_{π_{1}∘π_{2},ξ}^{q}.

It follows that

e(π_{1}) Sym_{κ}(Ω) ∘ e(π_{2}) Sym_{κ}(Ω) = e(π_{1} ∘ π_{2}) Sym_{κ}(Ω).

Finally it is easily checked that |ℚ| = λ; and it follows that V^{ℚ} ⊨ 2^{κ} = λ. This completes the proof of Lemma 2.7.

Summing up our work in this section, we obtain the following result.

Theorem 2.14. Let V be a transitive model of ZF C and let κ, λ be cardinals such that ω < κ^{<κ} = κ < λ = λ^{κ}. Let α be any ordinal such that α < λ^{+}. Then there exists a notion of forcing ℚ such that

(1) ℚ is κ-closed;

(2) ℚ has the κ^{+}- c.c.; and such that the following statements are true in the generic extension V^{ℚ}.

(a) 2^{κ} = λ.

(b) There exist groups H < G < Sym(κ) such that |H| = κ and the normaliser tower of H in G terminates in exactly α steps.

Proof. Let γ be an ordinal such that max{α, λ} ≤ γ < λ^{+}. Then |W_{γ}| = λ. Let ℚ be the notion of forcing obtained by applying Lemma 2.7 to W = W_{γ}. By Lemma 2.5, ℚ satisfies our requirements.

3. Closed groups of uncountable degree

In this section, we will prove Theorem 1.12. Let κ be a regular uncountable cardinal such that κ^{<κ} = κ, and let G be a group which satisfies the κ^{+}-compatibility condition. Let L be a first-order language consisting of κ binary relation symbols. The following notion of forcing ℙ is designed to adjoin a structure M of cardinality κ for the language L such that G ≃ Aut M. This is sufficient; for then we can use

THE AUTOMORPHISM TOWER PROBLEM REVISITED 17

one of the standard procedures to code M into a graph Γ of cardinality κ such that Aut Γ ≃ Aut M. ( Cf. Section 5.5 of Hodges [3].)

Definition 3.1. Suppose that L_{0} ⊆ L and that N is a structure for the language L_{0}. Then a restricted atomic type in the free variable v for the language L_{0} using parameters from N is a set t of formulas of the form R(v, a), where R ∈ L_{0} and a ∈ N. An element c ∈ N is said to realise t if N ⊨ φ[c] for every formula φ(v) ∈ t. If no element of N realises t, then t is said to be omitted in N. Notice that if t is omitted in N, then t is not the trivial restricted atomic type ∅.

Definition 3.2. Let ℙ be the notion of forcing consisting of the conditions

p = (H, π, N , T)

such that the following hold.

(a) H is a subgroup of G such that |H| < κ.

(b) There exists an ordinal 0 < δ < κ and a subset L(N) ∈ [L]^{<κ} such that N is a structure with universe δ for the language L(N).

(c) π : H → Aut N is a group homomorphism.

(d) T is a set of restricted atomic types in the free variable v for the language L(N) using parameters from N. Furthermore, |T | < κ; and each t ∈ T is omitted in N.

We set (H_{2}, π_{2}, N_{2}, T_{2}) ≤ (H_{1}, π_{1}, N_{1}, T_{1}) if and only if

(1) H_{1} ⩽ H_{2}.

(2) N_{1} is a substructure of N_{2}.

(3) For all h ∈ H_{1} and α ∈ N_{1}, π_{2}(h)(α) = π_{1}(h)(α).

(4) T_{1} ⊆ T_{2}.

It is clear that the components (H, π, N) in each condition p ∈ ℙ are designed to generically adjoin a structure M of cardinality κ for the language L, together with an embedding π^{∗} of G into Aut M. The set T of restricted atomic types is needed to kill off potential extra automorphisms g ∈ Aut M ∖ π^{∗}[G], and thus ensure that π^{∗} is surjective.

Lemma 3.3. ℙ is κ-closed.

Proof. This is clear.

Lemma 3.4. ℙ has the κ^{+}- c.c.

Proof. Suppose that p_{i} = (H_{i}, π_{i}, N_{i}, T_{i}) ∈ ℙ for i < κ^{+}. After passing to a suitable subsequence if necessary, we can suppose that the following conditions hold.

(1) There exists a fixed structure N such that N_{i} = N for all i < κ^{+}.

(2) There exists a fixed set of restricted atomic types T such that T_{i} = T for all i < κ^{+}.

(3) There is a fixed group H such that for each i < κ^{+}, there exists an isomorphism f_{i} : H ≃ H_{i}.

(4) For each i < κ^{+}, let ψ_{i} : H → Aut N be the embedding defined by ψ_{i} = π_{i}∘f_{i}. Then ψ_{i} = ψ_{j} for all i < j < κ^{+}.

Since G satisfies the κ^{+}-compatibility condition, there exist ordinals i < j < κ^{+} and a surjective homomorphism φ : ⟨H_{i}, H_{j}⟩ → H_{i} such that

(a) φ ∘ f_{j} = f_{i}; and

(b) φ ↾ H_{i} = id_{H_{i}}. Let ⟨H_{i}, H_{j}⟩ → Aut N be the homomorphism defined by π = π_{i}∘φ. Clearly π_{i} ⊆ π. Note that if x ∈ H_{j}, then

π_{i} ∘ φ(x) = π_{i} ∘ (φ ∘ f_{j}) ∘ f_{j}^{−1}(x)
= π_{i} ∘ f_{i} ∘ f_{j}^{−1}(x)
= π_{j}(x).

Thus we also have that π_{j} ⊆ π. Consequently, we can define a condition p ≤ p_{i}, p_{j} by

p = (⟨H_{i}, H_{j}⟩, π, N , T) .

Lemma 3.5. For each p = (H, π, N , T) ∈ ℙ, there exists p^{+} = (H, π^{+}, N^{+}, T) ≤ p such that π^{+} : H → Aut N^{+} is an embedding.

Proof. Let N^{+} be the structure for the language L(N) such that

(a) the universe of N^{+} is the disjoint union N ⊔ H;

(b) for each relation R ∈ L(N), R^{N^{+}} = R^{N}.

THE AUTOMORPHISM TOWER PROBLEM REVISITED 19

In particular, if x ∈ N^{+} ∖ N, then x only realises the trivial restricted atomic type ∅ over N. Hence none of the restricted atomic types in T is realised in N^{+}. Let π^{+} : H → Aut N^{+} be the embedding such that for each h ∈ H,

(i) π^{+}(h)(x) = π(h)(x) for all x ∈ N; and (ii) π^{+}(h)(x) = hx for all x ∈ H. Then p^{+} = (H, π^{+}, N^{+}, T) ≤ p.

There is a slight inaccuracy in the proof of Lemma 3.5, as the universe of N^{+} should really be an ordinal δ < κ. However, the proof can easily be repaired: simply replace N^{+} by a suitable isomorphic structure. Similar remarks apply to the proofs of Lemmas 3.6 and 3.8.

Lemma 3.6. For each a ∈ G,

D_{a} = {(H, π, N , T) | a ∈ H}

is a dense subset of ℙ.

Proof. Let a ∈ G and p = (H, π, N , T) ∈ ℙ. We can suppose that a ∉ H. Let H^{+} = ⟨H, a⟩. Let C = {g_{i} | i ∈ I} be a set of left coset representatives for H in H^{+}, chosen so that 1 ∈ C. Let N^{+} be the structure for the language L(N) such that

(a) the universe of N^{+} is the cartesian product C × N; and

(b) for each relation R ∈ L(N),

((g_{i}, x), (g_{j}, y)) ∈ R^{N^{+}} iff i = j and (x, y) ∈ R^{N} .

By identifying each x ∈ N with the element (1, x) ∈ N^{+}, we can regard N as a substructure of N^{+}. Once again, each element (g_{i}, x) ∈ N^{+} ∖ N only realises the trivial restricted atomic type ∅ over N; and hence none of the restricted atomic types in T is realised in N^{+}.

Define an action of H^{+} on N^{+} as follows. If g ∈ H^{+} and (g_{i}, x) ∈ N^{+}, then

g(g_{i}, x) = (g_{j}, π(h)(x)),

where j ∈ I and h ∈ H are such that gg_{i} = g_{j}h. It is easily checked that this action yields a homomorphism π^{+} : H^{+} → N^{+}; and that (H^{+}, π^{+}, N^{+}, T) ≤ p.

20 WINFRIED JUST, SAHARON SHELAH, AND SIMON THOMAS

Lemma 3.7. For each α < κ,

E_{α} = {(H, π, N , T) | α ∈ N }

is a dense subset of ℙ.

Proof. Left to the reader.

Let F ⊆ ℙ be a generic filter, and let V^{ℙ} = V [F] be the corresponding generic extension. Working within V^{ℙ}, define

M = ⋃{N | There exists p = (H, π, N , T) ∈ F }

and

π^{∗} = ⋃{π | There exists p = (H, π, N , T) ∈ F }.

Then M is a structure for L of cardinality κ, and π^{∗} is an embedding of G into Aut M. So the following lemma completes the proof of Theorem 1.12.

Lemma 3.8. π^{∗} : G → Aut M is a surjective homomorphism.

Proof. Suppose that g ∈ Aut M ∖ π^{∗}[G]. Let M̃, π̃ be the canonical ℙ-names for M and π^{∗}; and let g̃ be a ℙ-name for g. Then there exists a condition p ∈ F such that

p ⊩ g̃ ∈ Aut M̃ and g̃ ≠ π̃(h) for all h ∈ G.

Using the fact that ℙ is κ-closed, we can inductively construct a descending sequence of conditions p_{m} = (H_{m}, π_{m}, N_{m}, T_{m}) for m ∈ ω such that the following hold.

(a) p_{0} = p.

(b) For all x ∈ N_{m}, there exists y ∈ N_{m+1} such that p_{m+1} ⊩ g̃(x) = y.

(c) For all h ∈ H_{m}, there exists z ∈ N_{m+1} such that p_{m+1} ⊩ g̃(z) ≠ π_{m+1}(h)(z). Let q = (H, π, N , T) be the greatest lower bound of {p_{m} | m ∈ ω} in ℙ. Then q ≤ p, and there exists g^{∗} ∈ Aut N ∖ π[H] such that q ⊩ g̃ ↾ N = g^{∗}. Let N^{+} be the structure defined as follows.

(1) The universe of N^{+} is the disjoint union N ⊔ H.

(2) For each relation R ∈ L(N), R^{N^{+}} = R^{N}.

(3) For each x ∈ N, let R_{x} ∈ L ∖ L(N) be a new binary relation symbol. Then we set (h, y) ∈ R_{x}^{N^{+}} iff h ∈ H, y ∈ N and π(h)(x) = y.

THE AUTOMORPHISM TOWER PROBLEM REVISITED 21

Once again, it is clear that none of the restricted atomic types in T is realised in N^{+}. Let π^{+} : H → Sym N^{+} be the embedding such that

(i) π^{+}(h)(x) = π(h)(x) for all x ∈ N; and (ii) π^{+}(h)(x) = hx for all x ∈ H. Then it is easily checked that π^{+}[H] ⩽ Aut N^{+}. Finally let t be the restricted atomic type defined by

t = {R_{x}(v, g^{∗}(x) | x ∈ N };

and let T^{+} = T ∪ {t}.

Claim 3.9. t is omitted in N^{+}.

Proof of Claim 3.9. If z ∈ N^{+} realises t, then z = h ∈ H. And if x ∈ N, then N^{+} ⊨ R_{x}(h, g^{∗}(x)), and so π(h)(x) = g^{∗}(x). But this contradicts the fact that g^{∗} ∈ Aut N ∖ π[H].

Thus q^{+} = (H, π^{+}, N^{+}, T^{+}) ∈ ℙ. To simplify notation, suppose that q^{+} ∈ F; so that g^{∗} ⊆ g. Then for each x ∈ N, we have that M ⊨ R_{x}(1, x), and hence M ⊨ R_{x}(g(1), g^{∗}(x)). But this means that g(1) ∈ M realises t, which is the final contradiction.

4. τ_{κ} is increasing

In this section, we will complete the proof of Theorem 1.4. So let V ⊨ GCH and let κ, λ ∈ V be uncountable cardinals such that κ < cf(λ). Let α be any ordinal such that α < λ^{+}. It is well-known that there exists a centreless group T of cardinality κ such that Aut T = Inn T. (For example, we can take T = P GL(2, K), where K is a rigid field of cardinality κ. The existence of such a field follows from Lemma 1.7.) Thus we can assume that α ≥ 1. First consider the case when κ is a regular cardinal. Let ℚ be the notion of forcing which is given by Theorem 2.14. Then the following statements are true in the corresponding generic extension M = V^{ℚ}.

(a) κ^{<κ} = κ.

(b) 2^{κ} = λ.

(c) There exist groups H < G < Sym(κ) such that |H| = κ and the normaliser tower of H in G terminates in exactly α steps.

22 WINFRIED JUST, SAHARON SHELAH, AND SIMON THOMAS

Let ℙ ∈ M be the notion of forcing, given by Theorem 1.12, which adjoins a graph Γ of cardinality κ such that G ≃ Aut Γ. Then, applying Lemma 1.8, we find that the following statements are true in M^{ℙ}.

(1) 2^{κ} = λ.

(2) There exists a centreless group T of cardinality κ such that τ(T) = α.

Next suppose that κ is a singular cardinal. By the above argument, there is a generic extension V^{ℙ∗ℚ} in which the following statements are true.

(i) 2^{ω_{1}} = 2^{κ} = λ. (ii) There exists a centreless group T of cardinality ω_{1} such that τ(T) = α. Let G = T × Alt(κ). Clearly |G| = κ; and the following theorem implies that τ(G) = τ(T) = α. This completes the proof of Theorem 1.4.

Theorem 4.1. Suppose that ω ≤ θ < κ. If H is a centreless group of cardinality θ such that τ(H) ≥ 1, then τ(H × Alt(κ)) = τ(H).

Corollary 4.2. If ω ≤ θ < κ, then τ_{θ} ≤ τ_{κ}.

The remainder of this section will be devoted to the proof of Theorem 4.1. Let G = H × Alt(κ); and let H_{β}, G_{β} be the β^{th} groups in the automorphism towers of H, G respectively. We will eventually prove by induction that G_{β} = H_{β} × Sym(κ) for all β ≥ 1. To accomplish this, we need to keep track of φ[Alt(κ)] for each automorphism φ of G_{β}. The next lemma shows that for all φ ∈ Aut G_{β}, either φ[Alt(κ)] ⩽ H_{β} or φ[Alt(κ)] ⩽ Sym(κ). The main point will be to eliminate the possibility that φ[Alt(κ)] ⩽ H_{β}. This is straightforward when β is a successor ordinal. To deal with the case when β is a limit ordinal, we will make use of the result that Alt(κ) is strictly simple.

Lemma 4.3. Suppose that A is a simple nonabelian normal subgroup of the direct product H × S. Then either A ⩽ H or A ⩽ S.

Proof. Let 1 ≠ g = xy ∈ A, where x ∈ H and y ∈ S. If y = 1, then the conjugacy class g^{A} = x^{A} is contained in H, and so A = ⟨g^{A}⟩ ⩽ H. So suppose that y ≠ 1. Let π : H × S → S be the canonical projection map. Then 1 ≠ y ∈ π[A] ⩽ S and π[A] ≃ A. Hence there exists an element z ∈ π[A] ⩽ S such that zyz^{−1} ≠ y. Since

THE AUTOMORPHISM TOWER PROBLEM REVISITED 23

A ⊴ H × S, it follows that

1 ≠ zyz^{−1}y^{−1} = zxyz^{−1}y^{−1}x^{−1} = zgz^{−1}g^{−1} ∈ A ∩ S.

Arguing as above, we now obtain that A ⩽ S.

Definition 4.4. Let H be a subgroup of the group G. Then H is said to be an ascendant subgroup of G if there exist an ordinal γ and a set of subgroups {H_{β} | β ≤ γ} such that the following conditions are satisfied.

(a) H_{0} = H and H_{γ} = G.

(b) If β < γ, then H_{β} ⊴ H_{β+1}.

(c) If δ is a limit ordinal such that δ ≤ γ, then H_{δ} = ⋃_{β<δ} H_{β}.

Definition 4.5. A group A is strictly simple if it has no nontrivial proper ascendant subgroups.

Theorem 4.6 (Macpherson and Neumann [7]). For each κ ≥ ω, the alternating group Alt(κ) is strictly simple.

□

Lemma 4.7. Suppose that H is a centreless group. Let τ = τ(H) and let H_{τ} be the τ^{th} group in the automorphism tower of H. If A is a strictly simple normal subgroup of H_{τ}, then A ⩽ H_{0}.

Proof. Let β ≤ τ be the least ordinal such that A ⩽ H_{β}. First suppose that β is a limit ordinal. Then A = ⋃_{γ<β} (A ∩ H_{γ}), and A ∩ H_{γ} ⊴ A ∩ H_{γ+1} for all γ < β. Consequently, if γ < β is the least ordinal such that A ∩ H_{γ} ≠ 1, then A ∩ H_{γ} is a nontrivial ascendant subgroup of A. But this contradicts the assumption that A is strictly simple.

Next suppose that β = γ + 1 is a successor ordinal. Since A ∩ H_{γ} is a proper normal subgroup of A, it follows that A ∩ H_{γ} = 1. Now notice that A ⩽ H_{γ+1} ⩽ N_{H_{τ}} (H_{γ}) and H_{γ} ⩽ H_{τ} = N_{H_{τ}} (A), This implies that [A, H_{γ}] ⩽ A ∩ H_{γ} = 1. (For example, see Lemma 1.1.3 of Suzuki [12].) But then A ⩽ C_{H_{γ+1}}(H_{γ}) = 1, which is a contradiction. The only remaining possibility is that β = 0.

We will also make use of the well-known results that Aut(Alt(κ)) = Sym(κ) and Aut(Sym(κ)) = Sym(κ). (For example, see Theorem 11.4.8 of Scott [9].)

Proof of Theorem 4.1. Let τ = τ(H); and let

H = H_{0} ⊴ H_{1} ⊴ … H_{β} ⊴ H_{β+1} ⊴ … H_{τ} = H_{τ+1} = …

be the automorphism tower of of H. Let G = H × Alt(κ); and let

G = G_{0} ⊴ G_{1} ⊴ G_{2} ⊴ … G_{β} ⊴ G_{β+1} ⊴ …

be the automorphism tower of G. We will prove by induction on β ≥ 1 that G_{β} = H_{β} × Sym(κ).

First consider the case when β = 1. Let φ ∈ Aut G be any automorphism. Then φ[Alt(κ)] is a simple nonabelian normal subgroup of the direct product G = H × Alt(κ). By Lemma 4.3, either φ[Alt(κ)] ⩽ H or φ[Alt(κ)] ⩽ Alt(κ). Since |φ[Alt(κ)]| = κ > θ = |H|, it follows that φ[Alt(κ)] ⩽ Alt(κ). As φ[Alt(κ)] is a normal subgroup of G, we must have that φ[Alt(κ)] = Alt(κ). Note that C_{G}(Alt(κ)) = H. Hence we must also have that φ[H] = H. It follows that

G_{1} = Aut G = Aut H × Aut(Alt(κ)) = H_{1} × Sym(κ).

Next suppose that β = γ+1 and that G_{γ} = H_{γ} ×Sym(κ). Let φ ∈ Aut G_{γ} be any automorphism. By Lemma 4.3, either φ[Alt(κ)] ⩽ H_{γ} or φ[Alt(κ)] ⩽ Sym(κ). As Alt(κ) is a strictly simple group, Lemma 4.7 implies that φ[Alt(κ)] ⩽ Sym(κ). Since φ[Alt(κ)] is a simple normal subgroup of G_{γ}, it follows that φ[Alt(κ)] = Alt(κ). Using the facts that C_{G_{γ}} (Alt(κ)) = H_{γ} and C_{G_{γ}} (H_{γ}) = Sym(κ), we now see that φ[H_{γ}] = H_{γ} and φ[Sym(κ)] = Sym(κ). Hence

G_{γ+1} = Aut G_{γ} = Aut H_{γ} × Aut(Sym(κ)) = H_{γ+1} × Sym(κ).

Finally no difficulties arise when β is a limit ordinal.

References

[1] N. G. De Bruijn, Embedding theorems for infinite groups, Indag. Math. 19 (1957), 560–569.

[2] E. Fried and J. Kollár, Automorphism groups of fields, in Universal Algebra (E. T. Schmidt et al., eds.), Coloq. Math. Soc. Janos Boyali, vol 24, 1981, pp. 293–304.

[3] W. Hodges, Model theory, Encyclopedia of Mathematics and its Applications 42, Cambridge University Press, 1993.

[4] J. A. Hulse, Automorphism towers of polycyclic groups, J. Algebra 16 (1970), 347–398.

[5] D. W. Kueker, Definability, automorphisms and infinitary languages, in The Syntax and Semantics of Infinitary Languages (ed. J. Barwise), Lecture Notes in Math. 72, Springer, Berlin, 1968, pp. 152–165.

[6] K. Kunen, Set Theory, North-Holland, Amsterdam, 1980.

[7] H. D. Macpherson and P. M. Neumann, Subgroups of infinite symmetric groups, J. London Math. Soc. (2) 42 (1990), 64–84.

[8] A. Rae and J. E. Roseblade, Automorphism towers of extremal groups, Math. Z. 117 (1970), 70–75.

[9] W. R. Scott, Group Theory, Prentice-Hall, Englewood Cliffs, New Jersey, 1964.

[10] S. Shelah, First order theory of permutation groups, Israel J. Math. 14 (1973), 149–162.

[11] S. Solecki, Polish group topologies, preprint (1997).

[12] M. Suzuki, Group Theory II , Grundlehren der mathematischen Wissenschaften 248, Springer-Verlag 1986.

[13] S. Thomas, The automorphism tower problem, Proc. Amer. Math. Soc. 95 (1985), 166–168.

[14] S. Thomas, The automorphism tower problem II , to appear in Israel J. Math.

[15] H. Wielandt, Eine Verallgemeinerung der invarianten Untergruppen, Math. Z. 45 (1939), 209–244.

Department of Mathematics, Ohio University, Athens, Ohio 45701

Mathematics Department, The Hebrew University, Jerusalem, Israel

Mathematics Department, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019