On Minimal Artinian Modules and Minimal Artinian Linear Groups

The paper is devoted to the study of some important types of minimal artinian linear groups. The authors prove that in such classes of groups as hypercentral groups (so also, nilpotent and abelian groups) and FC-groups, minimal artinian linear groups have precisely the same structure as the corresponding irreducible linear groups. 2000 Mathematics Subject Classification. 20E36, 20F28. Let F be a field, A a vector space over F . The group GL(F,A) of all automorphisms of A and its distinct subgroups are the oldest subjects of investigation in Group Theory. For the case when A has a finite dimension over F , every element of GL(F,A) defines some nonsingular n×n-matrix over F , where n= dimF A. Thus, for the finitedimensional case, the theory of linear groups is exactly the theory of matrix groups. That is why the theory of finite-dimensional linear groups is one of the best developed in algebra. However, for the case when dimF A is infinite, the situation is totally different. The study of this case always requires some essential additional restrictions. Thus, the transition from the study of finite groups to the study of infinite groups generated the finiteness conditions. It is natural to apply these finiteness conditions to the study of infinite-dimensional linear groups. The study of finitary linear groups (the linear analogies of FC-groups) shows the effectiveness of such approach (cf. a survey of Phillips [6]). The groups having a finite composition series were one of the first generalization of the finite groups. Let G ≤ GL(F,A), then we can consider A as an FG-module. We say that A has a finite composition length if A has a finite series 〈0〉 = B0 ≤ B1 ≤ ··· ≤ Bn = A of FG-submodules, every factor of which is a simple FG-module. We can consider G/CG(Bi+1/Bi) as an irreducible linear group, 0 ≤ i ≤ n− 1. Let T = 0≤i≤n−1CG(Bi+1/Bi); then G/T ≤ X0≤i≤n−1G/CG(Bi+1/Bi), and T is a nilpotent bounded p-subgroup whenever charF = p, or T is a nilpotent divisible torsion-free subgroup whenever charF = 0. Thus, the case of irreducible linear groups is basic. Irreducible linear groups as the automorphism groups of abelian chief factors, play a crucial role in Group Theory, and their investigation is very useful for the solution of many group theoretical problems. For the infinite-dimensional case, the irreducible groups under some natural restrictions have been studied by Hartley and McDougall [2], Zăıcev [13], Robinson and Zhang [9], Franciosi, de Giovanni, and Kurdachenko [1], and Kurdachenko and Subbotin [5]. 708 L. A. KURDACHENKO AND I. YA. SUBBOTIN The minimal and the maximal conditions were the very next classical finiteness conditions that have appeared in algebra. Note that every FG-module of finite composition length is artinian (i.e., it satisfies the minimal condition on FG-submodules) and noetherian (i.e., it satisfies the maximal condition on FG-submodules). Let R be a ring, A an artinian R-module. Put Sicl(A)= { B | B is an R-submodule of A and has no finite composition series. (1) If A has no finite composition series, then Sicl(A) ≠ ∅. Since A is artinian, Sicl(A) has a minimal elementM . Thus, if U is a proper R-submodule ofM, then U has a finite composition length. An R-module M is said to be a minimal artinian, if M has no finite composition series, but each of its proper submodule has a finite composition length. Thus every artinian module includes a minimal artinian submodule. On the other hand, the structure of artinian modules depends on the structure of its minimal artinian submodules, therefore, the study of minimal artinian modules is one of the important steps for the study of artinian modules. Let again F be a field, A a vector space over F , G ≤ GL(F,A). We want to consider the situation when A is a minimal artinian FG-module. This consideration will lead us to the fact that the group G is lying in the class X such that all irreducible X-groups have been described. So we may set that if an FG-module B has finite composition series, then the structure of G is defined. Let F be a field, A a vector space over F , G ≤ GL(A). A group G is called a minimal artinian if the following conditions hold: (MA1) A has no finite composition series; (MA2) if B is a proper FG-submodule of A, then B has a finite composition length. The study of minimal artinian FG-modules (as any FG-module) consists of two parts: the study of internal structure of the module and the study of the group G/CG(A). The last group is imbedded in GL(F,A), that is, it is a linear minimal artinian group. Our paper is devoted to the study of some important types of minimal artinian linear groups. The main results of this paper show that in such classes of groups as hypercentral groups (so also, nilpotent and abelian groups) or FC-groups the minimal artinian linear groups have precisely the same structure as the corresponding irreducible linear groups have. Now we mention some needed results on hypercentral irreducible groups. The irreducible ZG-modules have been studied in [5]. These results can be extended almost without changes on the case of irreducible subgroups of GL(F,A), where A is a vector space over a field F . Lemma 1. Let F be a field, G a group, A a simple FG-module, I = AnnFG(A). If C/I is a center of FG/I, then C/I is an integral domain. In particular, the periodic part of ζ(G/CG(A)) is a locally cyclic p′-subgroup where p = charF . As usual, 0′ denotes the set of all primes. This statement is an immediate corollary of the known theorem of I. Schur. A groupG is said to have finite 0-rank r0(G)= r (or finite torsion-free rank) ifG has a finite subnormal series with exactly r infinite cyclic factors being the others periodic. ON MINIMAL ARTINIAN MODULES AND MINIMAL . . . 709 We note that every refinement of each of these series has only r infinite cyclic factors. Since every two subnormal series have the isomorphic refinements, 0-rank is independent of the choice of the subnormal series. Note also that if G is a locally nilpotent group of finite 0-rank, then the factor-group G/t(G) by the periodic part t(G) has a finite special rank. Lemma 2. Let G be a hypercentral group of finite 0-rank, F a locally finite field, A a simple FG-module. Then ζ(G/CG(A)) is periodic. This lemma follows from [4, Theorem 2]. Lemma 3. Let F be a field, p = charF , G an abelian group of finite 0-rank. (1) If the field F is locally finite, and G is a locally cyclic p′-group, then there exists a simple FG-module A such that CG(A)= 〈1〉. (2) If F is not locally finite, and t(G) is a locally cyclic p′-group, then there exists a simple FG-module A such that CG(A)= 〈1〉. This construction is contained in [2]. Lemma 4. Let F be a field, p = charF , G an abelian group of infinite 0-rank. If t(G) is a locally cyclic p′-group, then there exists a simple FG-module A such that CG(A)= 〈1〉. This assertion has been proved in [9] for the case of finite field, however it is valid also for an arbitrary field. Lemma 5. Let F be a field, p = charF , G a hypercentral group of finite 0-rank, C = ζ(G), T = t(C). (1) If the field F is locally finite, and C = T is a locally cyclic p′-group, then there exists a simple FG-module A such that CG(A)= 〈1〉. (2) If F is not locally finite and T is a locally cyclic p′-group, then there exists a simple FG-module A such that CG(A)= 〈1〉. Lemma 6. Let F be a field, p = charF , G a hypercentral group of infinite 0-rank, C = ζ(G), T = t(C). If T is a locally cyclic p′-group, then there exists a simple FGmodule A such that CG(A)= 〈1〉. The proof of both these assertions is similar to the proof of the respective results of [5]. Lemma 7. Let R be a ring, A a minimal artinian R-module. Then A does not decompose into a direct sum of two proper R-submodules. The lemma is obvious. If A is an R-module, then let SocR(A) denotes the sum of all minimal R-submodules whenever A includes such submodules, and SocR(A)= 〈0〉 otherwise. Clearly, SocR(A) is a direct sum of some minimal R-submodules (if it is nonzero). If A is an artinian R-module, then SocR(A)≠ 〈0〉 and SocR(A) is a direct sum of finitely many minimal R-submodules. So we come to the following lemma. Lemma 8. LetR be a ring,A aminimal artinianR-module. Then SocR(A) is a nonzero proper submodule of A. 710 L. A. KURDACHENKO AND I. YA. SUBBOTIN Lemma 9. Let F be a field, G a group, H a normal subgroup having a finite index in G, A an FG-module. If A has finite composition length as an FG-module, then A has finite composition length as an FH-module.