Milan J. Math. Vol. 77 (2010) 127–150 DOI 10.1007/s00032-014-0215-9 © 2014 Springer Basel

Milan Journal of Mathematics

On Some Variants of Theorems of Schur and Baer Martyn R. Dixon, Leonid A. Kurdachenko and Aleksander A. Pypka Abstract. A well-known theorem of I. Schur states that if G is a group and G/ζ(G) is finite then G is finite. We obtain an analogue of this, and theorems due to R. Baer and P. Hall, for groups G that have subgroups A of Aut(G) such that A/Inn(G) is finite. Mathematics Subject Classification (2010). Primary 20F14; Secondary 20F28. Keywords. central factor group,automorphism,hypercentre.

1. Introduction The relationship between the central factor group, G/ζ(G), of a group G and its derived subgroup, G , is quite intimate. This relationship was first studied by I. Schur [13] and one consequence of his results is the well-known fact that if G/ζ(G) is finite then G is also finite. In fact more can be said: if π is a set of primes and G/ζ(G) is a finite π-group then G is also a finite π-group (see [15], for example). For a group G we let ζn (G) denote the n th term of the upper central series and let γn (G) denote the n th term of the lower central series. In [1], R. Baer generalized Schur’s theorem. He proved that if the factor group G/ζk (G) is finite, then γk+1 (G) is also finite. In the opposite direction, P. Hall [3] proved that if γk+1 (G) is finite, then the factor group G/ζ2k (G) is finite. There are various other generalizations of Schur’s theorem, including [2] and [11]. Among these is a variation concerning automorphisms due to P. Hegarty [8], which we now describe. For a group G let Aut(G) denote the automorphism group of G and let A be a subgroup of Aut(G). As usual we let CG (A) = {g ∈ G|α(g) = g for each α ∈ A}, and [G, A] = g −1 α(g)|g ∈ G, α ∈ A.

M. R. Dixon, L. A. Kurdachenko and A. A. Pypka

In the case when A = Aut(G), Hegarty proved that if G/CG (Aut(G)) is finite, then [G, Aut(G)] is also finite. One very strong consequence of Hegarty’s result is that Aut(G) is then finite. Infinite groups with finite automorphism group have been studied by many authors. Clearly, when Aut(G) is finite, G/ζ(G) is also finite and hence G is likewise finite. Thus if G is torsionfree then G is abelian and in a series of papers [4, 9, 5, 6, 7] Hallett and Hirsch, and Hirsch and Zassenhaus, have given a description of the torsionfree groups with finite automorphism group. It is of some interest to examine the case when the subgroup A is a proper subgroup of Aut(G). In general CG (A) is not normal in G, although this is the case when Inn(G) ≤ A since then CG (A) ≤ CG (Inn(G)) = ζ(G). Clearly CG (A) is always A-invariant. On the other hand, the subgroup [G, A] is normal for every subgroup A of Aut(G), as is easily seen. For these reasons, we shall therefore restrict our discussion to subgroups A of Aut(G) such that Inn(G) ≤ A and obtain an analogue of Schur’s theorem in the case when A/Inn(G) is finite. Furthermore, for this case we also obtain generalizations of the theorems of Baer and Hall. Our first result in this direction is then as follows. Theorem 1. Let G be a group and let A be a subgroup of Aut(G). Suppose that Inn(G) ≤ A and that A/Inn(G) is finite of order k. If G/CG (A) is finite of order t, then [G, A] is finite of order at most ktd , where d = (1/2)(logp t + 1) and p is the least prime dividing t. We remark that if A is a group of automorphims of a group G and if X is an A-invariant normal subgroup of G then A/CA (X) is the group of automorphisms of X induced by A and likewise A/CA (G/X) is the group of automorphisms of G/X induced by A. If α ∈ A then we shall usually denote the automorphism induced by α on X or on G/X by α ¯ , it being clear from the context which meaning is meant. Thus α ¯ (x) = α(x), whenever x ∈ X and α ¯ (gX) = α(g)X, whenever g ∈ G. Starting with CG (A) and [G, A], where Inn G ≤ A, we can define the upper and lower A-central series of G. First we let ζ1 (G, A) = CG (A), a normal A-invariant subgroup of G. This allows us to define an ascending series of normal A-invariant subgroups of G, with terms ζν (G, A), where ζν+1 (G, A)/ζν (G, A) = ζ1 (G/ζν (G, A), A/CA (ζν (G, A)). We shall often abuse notation and just write this latter group as ζ1 (G/ζν (G, A), A). As usual, if ν is a limit ordinal, then we let ζν (G, A) = μ<ν ζμ (G, A). The last term ζ∞ (G, A) = ζγ (G, A) of this series is called the upper A-hypercentre of G and the ordinal γ is called the A-upper central length of G, which we denote by zl(G, A). Likewise, the lower A-central series of G is the descending, normal A-invariant series G = γ1 (G, A) ≥ γ2 (G, A) ≥ · · · ≥ γν (G, A) ≥ γν+1 (G, A) ≥ . . . defined by γ2 (G, A) = [G, A], and for each ordinal ν we have that γν+1 (G, A) = the notation slightly. As usual, for the limit ordinal λ, [γν (G, A), A], again abusing  we define γλ (G, A) = μ<λ γμ (G, A). The last term γδ (G, A) = γ∞ (G, A) is called

On Some Variants of Theorems of Schur and Baer

the lower A-hypocentre of G. Our result in this direction is as follows (and note that in Theorem 4 below we obtain better bounds for the order of γ∞ (G, A)). Theorem 2. Let G be a group and let A be a subgroup of Aut(G) such that Inn(G) ≤ A and |A : Inn(G)| = k is finite. Let Z be the upper A-hypercentre of G. Suppose that zl(G, A) = m is finite and that G/Z is finite of order t. Then γm+1 (G, A) is finite, and there exists a function β such that |γm+1 (G, A)| ≤ β(k, m, t). Our result analogous to Hall’s is as follows: Theorem 3. Let G be a group and let A be a subgroup of Aut (G) such that Inn(G) ≤ A and |A : Inn(G)| = k is finite. If γm+1 (G, A) is finite of order t for some positive integer m, then G/ζ2m (G, A) is finite of order at most η(k, m, t), for some function η. In Section 2 we give the proofs of these results. Our notation is that in standard use, when not explained. In particular, if x ∈ G, then we let τx denote the automorphism induced by the element x, so τx (g) = x−1 gx for each g ∈ G.

2. The proofs of the theorems Proof of Theorem 1. As we saw above CG (A) ≤ ζ(G), so G/ζ(G) is finite and has order at most t. Then G is finite and, by a theorem of J. Wiegold [14], |G | ≤ w(t) = tm , where m = (1/2)(logp t − 1) and p is the least prime dividing t. Let Gab = G/G . For each α ∈ A we let α ¯ : Gab −→ Gab denote the automorphism of Gab induced by α. The map α ¯ defines an endomorphism dα of Gab where dα (u) = u−1 α ¯ (u), for all u ∈ Gab . ¯ ], Ker dα = CGab (¯ α) and we have [Gab , α ¯] ∼ α). Clearly Im dα = [Gab , α = Gab /CGab (¯   α) and since CG (A) ≤ CG (α) we have |Gab /CGab (¯ α)| ≤ Of course CG (α)G /G ≤ CGab (¯ ¯ ]| ≤ t, for each α ∈ A. Let {α1 , . . . , αk } be a transversal t. It follows that |[Gab , α to Inn G in A. If β ∈ (Inn G)α then β = τz ◦ α for some element z ∈ G. Let g be an arbitrary element of G. Then [g, β] = g −1 z −1 α(g)z and hence [g, β]G = ¯ ] = [g, α]G so [Gab , α ¯] = [g, α]G , since G/G is abelian. On the other hand, [gG , α   [G, α]G /G . We have [G, A] = [G, β]|β ∈ A = [G, β]|β ∈ (Inn G)αi , 1 ≤ i ≤ k and   [G, αj ]G /G = [Gab , α ¯ j ]. [G, A]G /G = 1≤j≤k

|[G, A]G /G |

1≤j≤k

|G |

≤ tk. Since ≤ tm we have |[G, A]| ≤ tk · tm = It follows that  ktm+1 . Finally m + 1 = (1/2)(logp t + 1) so the result holds. Corollary 1. [8]. Let G be a group and suppose that G/CG (Aut(G)) is finite. Then [G, Aut(G)] is finite.

M. R. Dixon, L. A. Kurdachenko and A. A. Pypka

Proof of Theorem 2. Let 1 = Z0 ≤ Z1 ≤ · · · ≤ Zm−1 ≤ Zm = Z be the upper A-central series of G. We use induction on m. If m = 1, then Z1 = CG (A) has index at most t in G, and Theorem 1 implies that [G, A] = γ2 (G, A) is finite of order at most ktd(1) where d(1) = (1/2)(log2 t + 1). Suppose inductively that the result is true for some integer m − 1 and let G be a group satisfying the hypotheses of the theorem with zl(G, A) = m. Let ¯ be the naturally induced automorphim of L. The L = G/Z1 . Let α ∈ A and let α mapping η : A −→ Aut (L), defined by η(α) = α ¯ is a homomorphism such that η(Inn(G)) = Inn (L). Thus Inn(L) ≤ η(A) and η(A)/Inn(L) is finite of order at most k. Furthermore the series, 1 = Z1 /Z1 ≤ Z2 /Z1 ≤ · · · ≤ Zm−1 /Z1 ≤ Zm /Z1 is the upper A-central series of G/Z1 . Since zl(G/Z1 , η(A)) = m − 1, the induction hypothesis implies that γm (G/Z1 , η(A)) is finite and there is a function β(k, m−1, t) such that |γm (G/Z1 , η(A))| ≤ β(k, m − 1, t). As above we have γm (G/Z1 , A) = γm (G, A)Z1 /Z1 . Let K/Z1 = γm (G/Z1 , η(A)) = γm (G/Z1 , A). We now apply Theorem 1 to K. We note that K/Z1 is finite and |K/Z1 | ≤ β(k, m − 1, t) = r. We deduce from Theorem 1 that [K, A] is finite of order at most krd(m) where d(m) = (1/2)(log2 r + 1). Thus γm+1 (G, A) = [γm (G, A), A] ≤  [K, A], so that γm+1 (G, A) is finite and has order at most krd . The function β(k, m, t) is defined recursively by β(k, 1, t) = ktd(1) , where d(1) = 1/2(log2 t + 1), and β(k, m + 1, t) = k(β(k, m, t))d(m+1) , where d(m + 1) = (1/2)(log2 (β(k, m, t) + 1). This theorem implies that the lower A-hypocentre of G is finite and that its order is bounded by the function β(k, m, t), defined above. However, we can obtain better bounds for |γ∞ (G, A)|, independent of m and Theorem 4 below gives such bounds. The bound we obtain is simply dependent upon the order of the factor group of the upper A-hypercentre and |A : Inn(G)|. We shall require a couple of lemmas for the proof of Theorem 4 and also the following concepts. If G is a group, R is a ring and B an RG-module we construct the upper RG-central series of B, 0 = B0 ≤ B1 ≤ · · · ≤ Bα ≤ Bα+1 ≤ · · · ≤ Bγ , where B1 = ζRG (B) = {a ∈  B|a(g − 1) = 0 for all g ∈ G}, Bα+1 /Bα = ζRG (B/Bα ), for all ordinals α < γ, Bλ = α<λ Bα , for limit ordinals λ, and ζRG (B/Bγ ) = 0. The last term Bγ of this series is called the upper RG-hypercentre of B and is denoted ∞ (B) The ordinal γ is called the G-central length of B and will be denoted by by ζRG zlG (B). If B = Bγ , then B is called RG-hypercentral; if γ is finite, then B is called RG-nilpotent.

On Some Variants of Theorems of Schur and Baer

Let C, D be RG-submodules of B and suppose that D ≤ C. The factor C/D is called G-eccentric if CG (C/D) = G. The RG-submodule C is said to be RGhypereccentric if there is an ascending series 0 ≤ C0 ≤ C1 ≤ · · · ≤ Cα ≤ · · · ≤ Cγ = C of RG-submodules of C such that each factor Cα+1 /Cα is a G-eccentric simple RGmodule, for every α < γ. Following D. I. Zaitsev [16], we say that the RG-module B has the Z-decomposition, ∞ ∞ (B) ⊕ ERG (B), B = ζRG ∞ (B) is the maximal RG-hypereccentric RG-submodule of B. Note that where ERG ∞ (B) contains every RG-hypereccentric RGin this case it is easy to see that ERG ∞ (B) is the unique largest maximal RG-hypereccentric RGsubmodule. Thus ERG submodule of B.

Lemma 1. Let G be a finite nilpotent group and let B be a ZG-module. Suppose that B contains a ZG-nilpotent ZG-submodule C such that |B/C| = t is finite. Then B contains a finite ZG-submodule K of order at most t such that B/K is ZG-nilpotent. Proof. Since B/C is finite there is a finite subset M of B such that M ZG + C = B. Let V = M ZG, and U = C ∩ V . Then U is ZG-nilpotent and |V /U | = t. Since G is finite, V is a finitely generated abelian group and the natural semidirect product, V G, is abelian-by-finite. Of course U is also finitely generated, its torsion subgroup, T , is finite and U = T ⊕ W for some torsionfree subgroup W . Let Y = U |T | , a characteristic subgroup of U . Then Y is a ZG-submodule of B and we note that U/Y is finite. Since G is nilpotent, the finite factor module V /Y has a Z-decomposition, ∞ (V /Y ) and E/Y = E ∞ (V /Y ). by [16], say V /Y = X/Y ⊕ E/Y , where X/Y = ζZG ZG Now U/Y is ZG-nilpotent so U ≤ X. Also |V /U | = t, so |E/Y | ≤ t. The choice of E implies that E is a ZG-submodule of V , and hence the torsion subgroup K of E is also a ZG-submodule. Since Y is torsionfree, |K| ≤ |E/Y | ≤ t. Now E/K is Z-torsionfree and contains (Y + K)/K, a ZG-nilpotent submodule of finite index. It follows that E/K is also ZG-nilpotent. However B = V + C and C and V /K are ZG-nilpotent. This implies that B/K is also ZG-nilpotent, as required.  We need the following generalization of a theorem of L. A. Kaluzhnin [10]. Lemma 2. Let G be a group and let A be a subgroup of Aut(G) such that Inn(G) ≤ A. Suppose that G has a series of A-invariant subgroups 1 = Z0 ≤ Z1 ≤ · · · ≤ Zm whose factors are A-central. Then γm (G, A) ≤ CG (Zm ). We can now obtain the bound we have been seeking. Theorem 4. Let G be a group and let A be a subgroup of Aut(G) such that Inn(G) ≤ A and |A : Inn(G)| = k is finite. Let Z be the upper A-hypercentre of G. Suppose that zl(G, A) = m is finite and G/Z is finite of order t. Then there exists a function β1 such that |γ∞ (G, A)| ≤ β1 (k, t).

M. R. Dixon, L. A. Kurdachenko and A. A. Pypka

Proof. Let 1 = Z0 ≤ Z1 ≤ · · · ≤ Zm−1 ≤ Zm = Z

(1)

be the upper A-central series of G. Each subgroup Zj is G-invariant and Zj /Zj−1 is A-central, for 1 ≤ j ≤ m. Let E = CA (Z). By a result of L. A. Kaluzhnin [10], A/E is nilpotent of nilpotency class at most m − 1. Let C = CG (Z). Since Inn(G) ≤ A, Lemma 2 implies that γm (G, A) ≤ C, and hence G/C is also nilpotent of nilpotency class at most m − 1. Since |G/Z| ≤ t it follows also that C/(Z ∩ C) ∼ = CZ/Z is finite of order at most t. Also we have C ∩ Z ≤ CC (E) so C/CC (E) has order at most t. Moreover Inn(C) ≤ E and it is easy to check that |E/Inn(C)| ≤ k. As in the proof of Theorem 1 we have D = C  [C, E] has order at most tkw(t) = ktd+1 , where d + 1 = 1/2(log2 t + 1). Let ν : G −→ A be the homomorphism defined by ν(g) = τg , for every g ∈ G. Since Z ≤ CG (C), we have ν(Z) ≤ CA (C). Also, t = |G/Z| ≥ |ν(G)/ν(Z)| = |Inn(G)/ν(Z)|, and this together with |A : Inn(G)| = k implies that |A/CA (C)| ≤ kt. Since C is A-invariant, D is also A-invariant and C/D is abelian, so is a ZA-module. Of course [C, E] ≤ D so E ≤ CA (C/D). Since A/E is a finite nilpotent group, ∞ (C/D), and (C/D)/((C?Z)D/D) ∼ so is A/CA (C/D). Next, (C ∩ Z)D/D ≤ ζZA = C/(C?Z)D is finite of order at most t. By Lemma 1, C/D contains a finite ZAsubmodule V /D of order at most t such that the factor module (C/D)/(V /D) ∼ = C/V is ZA-nilpotent. It follows that there exists a positive integer n such that γn (C, A) ≤ V . As we saw above γm (G, A) ≤ C so γn+m (G, A) ≤ V and, in particular, γ∞ (G, A) ≤ V . Finally, |V | = |D| · |V /D| ≤ tkw(t)t = kw(t)t2 = ktd+2 , where d + 2 = (1/2)(log2 t − 1) + 2 = (1/2)(log2 t + 3). Hence γ∞ (G, A) has order at most  β1 (k, t) = ktd+2 as required. Finally we are ready to prove Theorem 3. Our proof is very much analogous to the proof of Hall’s theorem, given in [12, Theorem 4.25], but we reproduce the details here, keeping track of various bounds which arise. We use the notation established above. Lemma 3. Let G be a group and let A be a group of automorphisms of G. Let H = CG (γh+1 (G, A)) and let K = CG (γk+1 (G, A)). Then [H, K] ≤ ζh+k−1 (G, A), if h + k > 0. Furthermore, if a + b + c ≥ 2h − 1 then [γa+1 (H, A), γb+1 (H, A)] ≤ ζc (G, A). Proof. We will need a variant of the Hall-Witt identity. Let x, y ∈ G, and α ∈ A. It is easy to prove that [[x, y −1 ], α]y α([[y, α−1 ], x])[[x−1 , α]−1 , y]x = 1.

(2)

If M, N are normal A-invariant subgroups of G then it is clear from this identity that γ2 ([M, N ], A) ≤ [γ2 (M, A), N ][γ2 (N, A), M ] and a very simple induction argument

On Some Variants of Theorems of Schur and Baer

establishes the truth of the inclusion γr+1 ([M, N ], A) ≤



[γi+1 (M, A), γj+1 (N, A)].

(3)

i+j=r

We set r = h + k − 1 ≥ 0 and set i + j = r. If j < h then i = h + k − 1 − j ≥ k and [γi+1 (H, A), γj+1 (K, A)] ≤ [γk+1 (H, A), K] = 1 and by similar reasoning, if j ≥ h, we have [γi+1 (H, A), γj+1 (K, A)] = 1. The equation (3) implies that γh+k ([H, K], A) = 1 from which it follows that [H, K] ≤ ζh+k−1 (G, A). We also observe that (3) implies γc+1 ([γa+1 (H, A), γb+1 (H, A)], A)  γi+1 (γa+1 (H, A), A), γj+1 (γb+1 (H, A), A)] ≤

(4)

i+j=c

=



[γa+i+1 (H, A), γb+j+1 (H, A)]

i+j=c

Let a + b + c ≥ 2h − 1 and i + j = c. Then (a + i) + (b + j) ≥ 2h − 1 so either a + i ≥ h or b + j ≥ h. In either case we have [γa+i+1 (H, A), γb+j+1 (H, A)] ≤ [γh+1 (G, A), H] = 1 and the result now follows from (4).



We are now all set for the proof of Theorem 3. Again the proof is directly connected to the proof of [12, Theorem 4.25]. Proof of Theorem 3. We have that γm+1 (G, A) is finite of order t and we set C = CG (γm+1 (G, A)). We set, for each s such that 0 ≤ s ≤ m, Fs =

γm−s+1 (C, A) . γm−s+1 (C, A) ∩ ζm+s (C, A)

We note that F0 is a quotient of γm+1 (C, A) so F0 is finite of order at most t. We s prove by induction on s that Fs is finite of order at most t(k+t!) ; our previous remark establishes the case s = 0 and we suppose that the result we wish to prove is true for s. By Lemma 3 we have, setting c = m + s, b = 0, a = m − s − 1, [γm−s (C, A), C] ≤ ζm+s (G, A). Furthermore, if x ∈ γm−s (C, A) and y ∈ C, then [x, y] = x−1 xτy , where τy is the inner automorphism induced by y. Since Inn(G) ≤ A it follows that [γm−s (C, A), C] ≤ [γm−s (C, A), A] = γm−s+1 (C, A). Consequently, γm−s (C, A) γm−s+1 (C, A) ∩ ζm+s (G, A) is a central factor of C so, for each α ∈ A, the map Φα : γm−s (C, A) −→ Fs defined, for each x ∈ γm−s (C, A), by x −→ [x, α](γm−s+1 (C, A) ∩ ζm+s (G, A)),

M. R. Dixon, L. A. Kurdachenko and A. A. Pypka

is a homomorphism. Let the kernel of Φα be K(α). Then γm−s (C, A)/K(α) has order s at most |Fs |, which is at most t(k+t!) , by the induction hypothesis. Since γm+1 (G, A) has order t it follows that G/C has order at most t! and we let g1 , g2 , . . . , gr be a transversal to C in G, where r ≤ t!. We also let α1 , . . . , αk be a transversal to Inn(G) in A. Define   K(αj ). K(τgj ) ∩ K= 1≤j≤r

1≤j≤k

Since K ≤ γm−s (C, A) we have [K, C] ≤ [γm−s (C, A), C] ≤ ζm+s (G, A). Also [K, gj ] = [K, τgj ] ≤ [K(τgj , τgj ] ≤ ζm+s (G, A). Hence [K, G] ≤ ζm+s (G, A). Furthermore, [K, Inn(G)] = [K, G] ≤ ζm+s (G, A) and [K, αj ] ≤ [K(αj ), αj ] ≤ ζm+s (G, A). Hence [K, A] ≤ ζm+s (G, A) which implies that K ≤ ζm+s+1 (G, A). Consequently K ≤ ζm+s+1 (G, A) ∩ γm−s (C, A). It follows that Fs+1 is an image of γm−s (C, A)/K which is finite of order at most |Fs |k+t! and the induction step follows. m Now we note that Fm is of order at most t(k+t!) and is clearly isomorphic to Cζ2m (G, A)/ζ2m (G, A). Since G/C has order at most t! it follows that G/ζ2m (G, A) m  is of order at most t(k+t!) t! and the result follows. m

It is clear that the function η is defined by η(k, m, t) = t(k+t!) t!.

References [1] R. Baer, Endlichkeitskriterien f¨ ur Kommutatorgruppen, Math. Ann. 124 (1952), 161– 177. [2] M. De Falco, F. de Giovanni, C. Musella, and Y. P. Sysak, On the upper central series of infinite groups, Proc. Amer. Math. Soc. 139 (2011), no. 2, 385–389. [3] P. Hall, Finite-by-nilpotent groups, Proc. Cambridge Philos. Soc. 52 (1956), 611–616. [4] J. T. Hallett and K. A. Hirsch, Torsion-free groups having finite automorphism groups. I, J. Algebra 2 (1965), 287–298. [5]

, Groups of exponent 4 as automorphism groups, Math. Z. 117 (1970), 183–188.

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[8] P. Hegarty, The absolute centre of a group, J. Algebra 169 (1994), no. 3, 929–935. [9] K. A. Hirsch and H. Zassenhaus, Finite automorphism groups of torsion-free groups, J. London Math. Soc. 41 (1966), 545–549. ¨ [10] L. A. Kaluzhnin, Uber gewisse Beziehungen zwischen einer Gruppe und ihren Automorphismen, Bericht u ¨ber die Mathematiker-Tagung in Berlin, Januar, 1953, Deutscher Verlag der Wissenschaften, Berlin, 1953, pp. 164–172. [11] L. A. Kurdachenko, J. Otal, and I. Ya. Subbotin, On a generalization of Baer Theorem, Proc. Amer. Math. Soc. 141 (2013), no. 8, 2597–2602.

On Some Variants of Theorems of Schur and Baer [12] D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups vols. 1 and 2, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, Heidelberg, New York, 1972, Band 62 and 63. ¨ [13] I. Schur, Uber die Darstellung der Endlichen Gruppen durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 127 (1904), 20–50. [14] J. Wiegold, Multiplicators and groups with finite central factor-groups, Math. Z. 89 (1965), 345–347. [15] , The Schur multiplier: an elementary approach, Groups—St. Andrews 1981 (St. Andrews, 1981), London Math. Soc. Lecture Note Ser., vol. 71, Cambridge Univ. Press, Cambridge, 1982, pp. 137–154. [16] D. I. Zaitsev, Hypercyclic extensions of abelian groups, Groups defined by properties of a system of subgroups (Russian), Akad. Nauk Ukrain. SSR Inst. Mat., Kiev, 1979, pp. 16–37, 152. Martyn R. Dixon Department of Mathematics University of Alabama Tuscaloosa, AL 35487-0350 U.S.A. e-mail: [email protected] Leonid A. Kurdachenko and Aleksander A. Pypka Department of Algebra, Facultet of mathematic and mechanik National University of Dnepropetrovsk Gagarin prospect 72 Dnepropetrovsk 10, 49010 Ukraine e-mail: [email protected], [email protected] [email protected] Received: October 7, 2013.