Pacific Journal of Mathematics ARTINIAN, ALMOST ABELIAN GROUPS AND THEIR GROUPS OF AUTOMORPHISMS ANNEMARIE SCHLETTE Vol. 29, No. 2 June 1969 PACIFIC JOURNAL OF MATHEMATICS Vol. 29, No. 2, 1969 ARTINIAN, ALMOST ABELIAN GROUPS AND THEIR GROUPS OF AUTOMORPHISMS ANNEMARIE SCHLETTE The principal results of this paper are several characteri- zations of the class of artinian almost abelian groups. Several corollaries of the main characterization are given, one of which states that a torsion group G which has a finite group of auto- morphisms is finite, and an abelian torsion group which has a countable automorphism group is finite. In a concluding section, the class of artinian almost abelian groups and the class of finite groups are characterized by formal group-theore- tical properties. A group is called artinian, if the minimum condition is satisfied by its subgroups. The structure of an abelian, artinian group is completely determined. Every artinian group that is known so far is an extension of an abelian, artinian group by a finite group. An extension of an abelian group by a finite group is called almost abelian. It is not known whether there exist further artinian groups. For this reason it seems to be worth while to consider the structure of artinian and almost abelian groups and we will give some characterizations of them. We will prove the following MAIN THEOREM. The following properties of the group G are equivalent: (1) G is artinian and almost abelian. (2) (a) Every abelian subnormal subgroup of G is artinian; (b) Every infinite epimorphic image of G possesses an abelian subnormal subgroup, not 1; (c) G is a torsion group. (3) (a) Every abelian subgroup of G is artinian; (b) Every epimorphic image, not 1, of G possesses an almost abelian normal subgroup, not 1. (4) (a) G is a torsion group; (b) Every torsion group of automorphisms of G is artinian and almost abelian. ( 5) (a) G is a torsion group; (b) The central quotient group of G is artinian and almost abelian; (c) Primary elementary abelian groups of automorphisms of G are countable. Sections 1 to 4 deal with preliminary lemmas, some of which may 403 404 ANNEMARIE SCHLETTE be of independent interest. We show, for instance, that the central quotient group of G is artinian and almost abelian if, and only if, G is almost abelian with artinian commutator subgroup G' (Proposition 3.2). Section 5 is dedicated to the proof of the Main Theorem. A simple application of the Main Theorem yields a characterization of the artinian groups with finite central quotient group (Corollary 6.1) and of the finite groups (Corollary 6.2), both by their automorphism groups. The results of the Main Theorem and of its corollaries finally are used to characterize the class of all artinian and almost abelian groups on the one hand and the class of all finite groups on the other hand by merely formal group theoretical properties. This is done in §7 (Theorem 7.1 and Theorem 7.2). NOTATIONS. lG = center of the group G, c U = centralizer of U in G, G n U = normalizer of U in G, G IG:U] = index of U in G, \G = intersection of all subgroups U of G with finite [G: U], {•••} = subgroup generated by the enclosed elements, xoy — χ~ιχy — χ-ιy-^χy for elements x, y in G, Jo Y = {χoy with x in X, y in Y) , Q o G = Gr = G1 = commutator subgroup of G, Q(n + 1) _ Q(n) Q(n) o G = set of all ^-elements in G for p a prime, p Gp = set of all gp with g in G [if G is multiplicatively written], pA = set of all pa with a in A [if A is additively written]. DEFINITIONS. A factor of G is an epimorphic image of a subgroup of G. The socle of an abelian p-group is the set of its elements of order p or 1. The group G is called artinian, if its subgroups satisfy the minimum condition; almost abelian, if it possesses an abelian [characteristic] subgroup of finite index; soluble, if G{n) = 1 for almost all n; locally finite, if every finitely generated subgroup of G is finite; hyper (almost) abelian, if every epimorphic image, not 1, of G possesses an (almost) abelian normal subgroup, not 1. hyperabelian ] aaG = product of all almost abelian [ normal subgroups of G. IfG locally finite J ARTINIAN, ALMOST ABELIAN GROUPS 405 An abelian group A is called divisible (radicable), if pA = A (Ap = A) for all primes p, and reduced, if it possesses no divisible (radicable) subgroup ^ 0 (Φ 1). 1Φ Lemmas on abelian groups* In this chapter we consider (additively written) abelian groups A. Any homomorphism σ of A maps the element x in A upon w. LEMMA 1.1. A reduced abelian p-group A with finite A/pA is finite. Proof. Since A/pA is finite, there exists a finitely generated subgroup F of A with A = F + pA. Since finitely generated, abelian p-groups are finite, F is finite. Therefore there exists a positive integer n with p".F — 0. Hence pnA = pnF + pn+ιA = p(pnA) is divisible. But A is reduced, so that pnA — 0. From A — F + pA we deduce A = F + [pi7 + p2A] = F + p2A= ... = = .F + p"A = F , so that A is finite. LEMMA 1.2. // A is an abelian, reduced, infinite p-group and if B is an abelian p-group Φ 0, then Horn (A, B) contains an uncount- ably infinite, elementary abelian p-subgroup. Proof. Since A is an abelian, reduced, infinite p-group, A/pA is infinite, by Lemma 1.1. Since B Φ 0, it contains a subgroup Z of order p. Now Horn (A, Z) S Horn (A, B) and Horn (A, Z) s Horn (A/pA, Z) , which is an uncountably infinite, elementary abelian p-group, (cf. [5], p. 206; E, and p. 208; Lemma 54.2). LEMMA 1.3. (a) If A and B are abelian, artinian groups, then the torsion subgroup of Horn (A, B) is finite. (b) Torsion groups of automorphisms of an abelian, artinian group are finite. 406 ANNEMARIE SCHLETTE Proof. The abelian artinian group A is a direct sum of a finite group F and a divisible group D, (cf. [5], p. 65; Th. 19.2). Hence there exists a positive integer n with nF = 0. Let X be the subgroup of all x in B with nx = 0. Since 5 is abelian and artinian, it is a consequence of ([5], p. 65; Th. 19.2) that X is finite. If σ is a torsion element of order i in Horn (A, B), then - (iZ>)σ = D(iσ) = DO = 0 and = {nF@nD)σ = w(Z>σ) = 0 so that A<7 s X. Therefore the torsion subgroup of Horn (A, B) is essentially the same as a group of homomorphisms of the finite group A/D ~ F into the finite group X, which implies its finiteness. This proves (a), and (b) is a consequence of (a) if we let A = B. LEMMA 1.4. // the abelian group A is the direct sum of in- finitely many subgroups A» Φ 0 , then A possesses an uncountable infinite, elementary abelian 2-group of automorphisms. The simple proof of this lemma may be left to the reader. 2* Stabilizing automorphism groups* PROPOSITION 2.1. Suppose B is a normal subgroup of the group G and A is a subgroup of G with AξΞ=B Γ) %G. Let Σ = Σ (A, B) G be the group of all automorphisms σ of G which fix every element in G/A and in B. Then Σ(A, B) = Horn (G/B, A) ~ Horn (G/BG', A) . β Proof. If σ is an automorphism in Σ, then a — 1 is a single valued mapping of G into A. If x and y are elements in G, then (xy)σ~ι — ^T/V^ctr1 = x'y'^x-1 = x^y*-1 , since y°~γ belongs to 4g%G. Therefore σ — 1 is a homomorphism of G into A whose kernel contains B. Thus y (a ) If σ is an element in Σ, then σ — 1 induces a homomorphism σ* of GjB into A with (Bgy - g*-1 . ARTINIAN, ALMOST ABELIAN GROUPS 407 Let a, β be two automorphisms in Σ and g be any element in G. Then aβ-i g"β(g-ψgβg-i = (g^ψgβ^ = g*~ψ->- , g = since g"'1 belongs to A § B and every element in B is a fixed element of β. Therefore (aβ)* = α* + β* for a, β in Σ , so that σ —> α* is a homomorphism of 2" into Hom (G/ί?, A). If <τ* = 0 and g is an element in G, then 1 - (Bgr = g-1 . This implies g = gσ for every g in G, so that σ == 1. Hence (b) The mapping σ —> σ* is a monomorphism of Σ into Hom (G/J5, A). For every 7 in Hom (G/B, A) define the mapping d of G into (? by the rule gδ = (Bflf)rflf for every g in G. This is clearly a single valued mapping of G into itself. If x,y are two elements in G, then (a?2/)β - (BxyYxy = (Bx)r(ByYxy = {Bx)'x(ByYy = a V since (S?/)r belongs to A<Ξ^$G. Thus δ is an endomorphism of G. If xδ = 1, then (βχ)ra; = 1, so that # belongs to 2? (since (ίta)r is in i g ΰ ). This implies (Bx)7 = Br = 1 so that x = 1. Hence <? is a monomorphism of G into G. Because of Br = 1 and (J?flf)r e A it is clear that <? induces the identity in both B and G/A. Now let # be any element in G and let x = Then xs = (BxYx - [B(βff-Ύ9\7(BgrιYg = (Bflf)i(Bflf)3']-1 fir = flf since (Bg~ιY belongs to A g J5. This shows that δ is an automorphism of G which belongs to Σ. Finally gδ~ι = (β^)r for every # in G , shows <ϊ* = 7 and (c) σ —> σ* is an isomorphism of Σ onto Hom (G/B, A) . The isomorphy of Hom (G/B, A) and Hom (G/BG', A) is an im- mediate consequence of the commutativity of A. 408 ANNEMARIE SCHLETTE LEMMA 2.2. // the group G possesses a characteristic subgroup A such that the torsion groups of automorphisms of A and of G/A are finite, then every torsion group Γ of automorphisms of G is almost abelian: The subgroup A of Γ which fixes every element in A and in G/A is abelian and has finite index in Γ. Proof. Let Γ be a torsion group of automorphisms of G. Denote by Γ resp. Γ the set of all those automorphisms in Γ, which induce A GjA the identity in A and in G/A respectively. Since A is characteristic in G, the sets Γ and Γ are both normal subgroups of Γ. Now A GίA Γ/Γ resp. Γ/Γ is essentially the same as the group of automorphisms A GίA of A resp. G/A, induced by Γ. Therefore it is (essentially) a torsion group of automorphisms of A resp. G/A, and hence, by hypothesis, finite. If A — Γ Π Γ , then A is a normal subgroup of Γ with finite A GlA Γ/A. By its definition, A is a subgroup of the stabilizer of the char- acteristic subgroup A of G. As such it is known to be abelian, see ([7], p. 88; Satz 19). This proves Lemma 2.2. 3* Central quotient group and commutator subgroup* In this chapter we aim at characterizing the groups with almost abelian, ar- tinian central quotient group (Proposition 3.2). LEMMA 3.1. If G is a group with abelian, radicable, not torsion- free central quotient group G/%G, then G is abelian. Proof. Suppose that G is not abelian. Then G/%G Φ 1 and since it is not torsionfree there exists a subgroup W/$G in G/ιG of order a prime p. Now Gf S gG c W, so that W is normal in G. Since G/&G is abe- lian, it follows that W<>GξΞ:$G and that w(w o g) = (w o g)w for every w e W, g e G . Since wp also belongs to 3G, it is centralized by every element in G. Therefore it follows from w° = w(wog) that Wp = (Wp)9 = (Wg)p = {w(Wog)Y = Wp(Wogy . Hence (w o g)p z= 1 for every w e W, g eG . By complete induction we get w9<L — w(w o gy for every positive integer i and in particular wgP = w(w°g)p = w , ARTINIAN, ALMOST ABELIAN GROUPS 409 which shows G'G* ^ G>&G Q cW . σ Therefore the epimorphic image G/c W of G/&G is an elementary abelian G 2>-group. Since elementary abelian p-groups Φ\ possess finite non- trivial epimorphic images, G/c W is equal to 1, because G/jG is radica- G ble. Hence c W = G, implying W ^ jGc W, a contradiction. There- G fore G is abelian. DEFINITION. fG = intersection of all subgroups U of G with finite index [G: U]. This is a well determined characteristic subgroup of G. PROPOSITION 3.2. The following properties of the group G are equivalent: ( i ) G/$G is artinian and almost abelian. (ii) If F is the uniquely determined characteristic subgroup of G with §G g F and FfoG = f(G/sG), then F is abelian, G/F is finite, and GoF is artinian. (iii) There exists an abelian normal subgroup A of G with finite G/A and artinian Go A. (iv) G is almost abelian and Gf is artinian. Proof, (i) —• (ii). Since G\ιG is artinian and almost abelian, ap- plication of ([1], pp. 3, 4, 22(B)) on G\%G shows (a) G/F= (G/sG)/f(G/8G) is finite and f(G/jG) = FfoG does not pos- sess finite epimorphic images. F/%G is abelian because Gj%G is almost abelian. Hence F' £ gG £ ιF g F and therefore (b) FJiF is an abelian radicable torsion group. Apply Lemma 3.1 on F to see (c) F is abelian. Let Γ be the group of automorphisms which is induced in F by G. Then Γ = G/c F is finite because FQc F by (c), and G/F is finite, G G by (a). Every automorphism 7 in Γ fixes every element in jG. Hence 7 — 1 is an endomorphism of the abelian group F whose kernel con- tains 3G. Consequently Fy~ι is an epimorphic image of the artinian group F/$G and as such Fr~ι is artinian. Since Γ is finite and F is abelian, F™ = Π Fr~ι is likewise artinian. From the definition of Γ, GoF= F^1 so that 410 ANNEMARIE SCHLETTE (d) GoF is artinian. (ii) — (iii). Let A = F. (iii) —» (iv). It is an immediate consequence of (iii) that G is al- most abelian. Since A is normal in G, so is G°A. Denote by σ the canonical epimorphism of G upon H = G/(GoA). Then B = Aσ is an abelian normal subgroup of H with finite H/B, as G/A is finite. From HoB = (GoA)σ = 1 we get βg giϊ, so that ίf/sίf is finite. This im- plies the finiteness of H', see ([6], p. 443; 15.1.13). But H' = Gfa~ G'/(G°A), so that G' is artinian as an extension of the artinian group Go A by the finite group H'. (iv) —• (iii). Obvious. (iii) ~>( i ). The group Γ of automorphisms of A which are in- duced in A by G is isomorphic to G/c A and it is finite since A Q c A G G and G/A is finite. If 7 is an automorphism in Γ, then 7 — 1 is an endomorphism of the abelian group A with Aγ~~ι QGoA. It follows that Ar~ι is artinian and A7"1 ~ A/K(y), where K(y) denotes the kernel of 7, i.e., K(y) = c g if # induces 7. Now A A Π δG = Π -BΓ(7) . Therefore A/(A Π sG) is isomorphic to a subgroup of the direct product of the finitely many artinian groups A/K(y) and as such A/(A Π jG) ~ A$G/iG is itself artinian. G/A%G is finite as an epimorphic image of G/A. Finally G/%G is an extension of the abelian artinian group A$G/%G by the finite group G/A%G. Hence G/%G is artinian and almost abelian. 4* Products of normal subgroups* In this section we shall establish a few lemmas on products of normal subgroups, which are needed later. In this context see also Corollary 6.4. DEFINITION. §aG = product of all hyperabelian normal subgroups of G. This is a well determined characteristic subgroup of G. LEMMA 4.1. // every abelian subnormal subgroup of the torsion group G is artinian, then (a) §aG is artinian and soluble; (b) §aG contains every hyperabelian subnormal subgroup of G; ( c) 1 is the only hyperabelian subnormal subgroup of G/ί)αG; (d) Proof. If S is a subnormal subgroup of G, then every abelian subnormal subgroup of §aS is a subnormal subgroup of G and hence, by hypothesis, artinian. Nontrivial epimorphic images of fyaS possess ARTINIAN, ALMOST ABELIAN GROUPS 411 by definition hyperabelian normal subgroups, not 1, and consequently abelian subgroups, not 1; and §aS is a torsion group, as is G. Apply ([4], p. 345, Satz 6.1, equivalence of (1) and (2)) to §aS to show that (e) §aS is artinian and soluble for every subnormal subgroup S of G. This implies (a) in the particular case S = G. Since S is subnormal in G, there exists a finite chain of subgroups Si of G with S = So, Si normal in S , S = G . i+1 n If S is hyperabelian, then S = §aS. By (e), fyaSi is a soluble, hence 0 hyperabelian characteristic subgroup of the normal subgroup S of t S . Hence fyaSi £ ϊjα£ , showing i+1 ί+1 S = ^αS £ £ ϊjα£ = £)cιG , 0 n so that §ctG contains every hyperabelian subnormal subgroup of G, proving (b). Now let T = S/§aG be a hyperabelian subnormal sub- group of G/^aG. Then £ is as an extension of the, by (a), soluble group ί)αG by the hyperabelian group T hyperabelian. Since £ is sub- normal in G, it follows from (b) that SξΞ fyaG. Hence T = 1, proving (c). (d) is an immediate consequence of (c). LEMMA 4.2. If N is a product of minimal normal subgroups of the group G, then N is a direct product of minimal normal sub- groups of G. This is proven by the customary lattice theoretical arguments; see for instance ([8], p. 208; Basis Theorem of Lattice Theory). LEMMA 4.3. If a minimal normal subgroup M of G is contain- ed in a product N of finite normal subgroups of G, then M is finite. Proof. Let t Φ 1 be an element in M. Since t belongs to the product N of finite normal subgroups of G, there exists a finite number n of finite normal subgroups F of G with JP< £ N and t tefLFi = F. Clearly F is a finite normal subgroup of G with t in F £ N. Hence lφMf]F and because of the minimality of M we have M=Mf]F^ F, so that M is finite. LEMMA 4.4. If 1 is the only abelian normal subgroup of G, then the following properties of the product N of finite normal subgroups