Serial group rings of finite groups. General linear and close groups

Serial group rings of finite groups. General linear and close groups

For a givenp, we determine when thepmodulargroup ring of a group from GL(n,q), SL(n,q) and PSL(n,q)-seriesis serial.

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Дата:2015
Автори: Kukharev, A., Puninski, G.
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Опубліковано: Інститут прикладної математики і механіки НАН України 2015
Назва видання:Algebra and Discrete Mathematics
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Цитувати:Serial group rings of finite groups. General linear and close groups / A. Kukharev, G. Puninski // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 1. — С. 115-125. — Бібліогр.: 22 назв. — англ.

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spelling nasplib_isofts_kiev_ua-123456789-1580032025-02-10T00:33:10Z Serial group rings of finite groups. General linear and close groups Kukharev, A. Puninski, G. For a givenp, we determine when thepmodulargroup ring of a group from GL(n,q), SL(n,q) and PSL(n,q)-seriesis serial. The authors are grateful to Alexandre Zalesski for a helpful discussion. The researchof the first author was supported by BRFFI grant F15RM-025. 2015 Article Serial group rings of finite groups. General linear and close groups / A. Kukharev, G. Puninski // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 1. — С. 115-125. — Бібліогр.: 22 назв. — англ. 1726-3255 2010 MSC:20C05,20G40. https://nasplib.isofts.kiev.ua/handle/123456789/158003 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description For a givenp, we determine when thepmodulargroup ring of a group from GL(n,q), SL(n,q) and PSL(n,q)-seriesis serial.
format Article
author Kukharev, A.
Puninski, G.
spellingShingle Kukharev, A.
Puninski, G.
Serial group rings of finite groups. General linear and close groups
Algebra and Discrete Mathematics
author_facet Kukharev, A.
Puninski, G.
author_sort Kukharev, A.
title Serial group rings of finite groups. General linear and close groups
title_short Serial group rings of finite groups. General linear and close groups
title_full Serial group rings of finite groups. General linear and close groups
title_fullStr Serial group rings of finite groups. General linear and close groups
title_full_unstemmed Serial group rings of finite groups. General linear and close groups
title_sort serial group rings of finite groups. general linear and close groups
publisher Інститут прикладної математики і механіки НАН України
publishDate 2015
url https://nasplib.isofts.kiev.ua/handle/123456789/158003
citation_txt Serial group rings of finite groups. General linear and close groups / A. Kukharev, G. Puninski // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 1. — С. 115-125. — Бібліогр.: 22 назв. — англ.
series Algebra and Discrete Mathematics
work_keys_str_mv AT kukhareva serialgroupringsoffinitegroupsgenerallinearandclosegroups
AT puninskig serialgroupringsoffinitegroupsgenerallinearandclosegroups
first_indexed 2025-12-02T04:57:21Z
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fulltext Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 20 (2015). Number 1, pp. 115–125 © Journal “Algebra and Discrete Mathematics” Serial group rings of finite groups. General linear and close groups Andrei Kukharev and Gena Puninski Communicated by V. V. Kirichenko Abstract. For a given p, we determine when the p-modular group ring of a group from GL(n, q), SL(n, q) and PSL(n, q)-series is serial. Introduction There is a recent progress in classifying finite groups G whose group ring FG over a modular field F is serial. It is shown in [15] that the crucial point in this description is making a list of simple finite groups (and fields of finite characteristics) with this property. For instance in [14] such a classification is given for symmetric and alternating groups; and [15] provides a list of sporadic simple groups and simple Suzuki groups with this property. Furthermore the first author described in [12] groups in the PSL(2, q)-series whose modular group rings are serial. In this paper we will continue this line of research by including into considerations all projective special linear groups PSL(n, q). Despite these groups are the main target of this paper, we have to make a bypass by considering general linear groups GL(n, q), and also special linear The authors are grateful to Alexandre Zalesski for a helpful discussion. The research of the first author was supported by BRFFI grant F15RM-025. 2010 MSC: 20C05,20G40. Key words and phrases: serial ring, group ring, general linear group, special linear group, projective special linear group. 116 Serial group rings of finite groups groups SL(n, q). The reason for such a detour is that for general linear groups the structure of Brauer trees of blocks is best known, due to results of Fong and Srinivasan [8,9]. Namely it is shown there that the Brauer tree of any block of GL(n, q) is an interval whose exceptional vertex is located at its end. From general theory it is known (see [1, Sect. 5]) that a block B of a group algebra is serial if and only if its Brauer tree is a star with the exceptional vertex at the center. Thus in the case of the serial p- modular group ring of GL(n, q) we obtain that all Brauer trees of blocks are intervals with at most two edges and, if a tree has two edges, then the exceptional vertex should have multiplicity one. Furthermore the number of edges in a particular block can be calculated using centralizers and normalizers of defect subgroups. There are rather few cases which are left to analyze, which is achieved in this paper without difficulty. In most cases descending from GL(n, q) to SL(n, q) and then to PSL(n, q) is a straightforward normal subgroup business, the only diffi- culty is when p divides q − 1. In this case more groups with serial group rings occur, and our analysis is based on [12] or directly by looking at character tables. There is no doubt that a similar approach applies to all classical groups but, because a myriad of details should be taken into account, we will postpone this to a future paper. 1. Preliminaries Recall that a module M over a ring R is said to be uniserial, if all submodules of M are linearly ordered by inclusion; and M is serial if it is a direct sum of uniserial modules. Furthermore R is called a serial ring, if R is serial as a right and left module over itself. It is known (see [2, Sect. 32]) that R is serial if and only if there exists a collection e1, . . . , en of orthogonal idempotents such that each right module eiR is serial, and the same is true for each left module Rej . For a general theory of serial rings the reader is referred to [19] or recent [4]. Within the class of artinian algebras over a field, the serial rings are also known as Nakayma algebras - see [3, Sect. 4.2]. Let G be a finite group and let F be a field of finite characteristic p. If p does not divide the order of G then, by Maschke’s theorem, the ring FG is semisimple artinian, hence serial. In this paper we will always assume that p divides |G|. A. Kukharev, G. Puninski 117 Let P denote a p-Sylow subgroup of G. Since (see [2, Theorem 32.3]) artinian serial rings are of finite representation type, it follows from Higman [10] that, if FG is serial, then P is a cyclic group. This gives a necessary condition for seriality, which is not always sufficient: for instance (see [1, p. 123]) the group SL(2, 5) for p = 5 gives a counterexample. Furthermore, the seriality of the group ring FG depends on charac- teristic of F only [6,16]. Thus in this paper (to ease references) we will always assume that F is algebraically closed. For instance, it is known (see [18,20] or [13]) that a p-modular group ring of a p-solvable group is serial. We say that the Brauer tree of a block is a star if it has no path of length more than 2. Here is a typical shape of a star with the exceptional vertex in the center: ◦ ◦ ◦ • ◦ ◦ ◦ A useful criterion for checking seriality is given by the following. Fact 1 (see [1, Sect. 5] or [7, Corollary VII.2.22]). A modular group ring R = FG is serial if and only if for each block B of R its Brauer tree is a star whose exceptional vertex (if any) is located in the center. Thus a satisfactory description of groups with serial group rings depends on the supply of information on Brauer trees of blocks, which is not always readily available. In some cases the seriality can be lifted from normal subgroups. Suppose that B is a block of the group algebra FG; H is a normal subgroup of G and b is a block of FH. A definition of the notion that B covers b can be found in [1, Sect. 14]. For instance if H contains a p-Sylow subgroup of B, then the principal block B0 of G covers the principal block b0 of H. Fact 2 (see [7, Theorem 6.2.7]). 1) Suppose that a block B of G covers a block b of H where H contains a defect group of B. Then B is serial if and only if b is serial. 2) Suppose that F is a field of characteristic p and let H be a normal subgroup of G whose index |G/H| is coprime to p. Then the ring FG is serial if and only if FH is serial. 118 Serial group rings of finite groups Suppose that B is a block of a modular group ring FG with a cyclic defect group D and let e denote the number of edges in the Brauer tree of B. For instance the defect group of the principal block B0 equals P . By CG(D) we denote the centralizer of D in G; and NG(D) is the normalizer of D. Fact 3 (see [1, Sect. 5, Theorem 1]). The number of edges e in the Brauer tree of a block B divides the order of the factor group NG(D)/CG(D), hence divides p−1. Furthermore the multiplicity of the exceptional vertex equals (|D| − 1)/e. For the principal block B0 the number of edges e equals to the order |NG(P )/CG(P )|. We will need one more technical result. Recall that Op′ denotes the largest normal subgroup ofG consisting of elements whose order is coprime to p. We say that an element g ∈ G is in the kernel of a block B if g acts trivially on every indecomposable projective module in B. Fact 4 (see [7, Lemma IV.4.12]). The kernel of the principal block of G equals Op′ . 2. General linear group In this section we will describe serial rings of general linear groups GL(n, q) over finite fields with q elements. Theorem 1. Let G = GL(n, q), n > 2 and let F be a field of characteristic p dividing the order of G. Then the group ring FG is serial if and only if one of the following holds. 1) n = 2 and p = q equal 2 or 3. 2) n = 2, 3, p = 3 and q ≡ 2, 5 (mod 9). For instance GL(3, 2) ∼= PSL(2, 7) and, for any field of characteristic 3, the group ring of this group is serial. Recall that the order of GL(n, q) equals qn(n−1)/2 · (q−1) · . . . · (qn −1). Thus if p divides the order of G, then either p | q or p divides qk − 1 for some k = 1, . . . , n. We will divide the proof of Theorem 1 in two parts. The case of the defining characteristic p | q is easy. Lemma 1. Let q = pr, G = GL(n, q) and F is a field of characteristic p. The group ring FG is serial if and only if n = 2, r = 1 and p equals 2 or 3. A. Kukharev, G. Puninski 119 Proof. If n = 3 then the matrices ( 1 1 0 0 1 0 0 0 1 ) and ( 1 0 0 0 1 0 0 1 1 ) generate a subgroup Cp × Cp, hence P is not cyclic; and we argue similarly for n > 4. Thus it remains to consider the case n = 2. If r > 2, it is easily checked that P is not cyclic, hence we may assume that p = q. Because p− 1, the index of SL(2, p) in GL(2, p), is coprime to p, it follows from Fact 2 that the seriality of group rings of GL(2, p) and SL(2, p) is equivalent. If p > 5 we conclude from [1, p. 124] that the Brauer tree of the principal block B0 of the group H = SL(2, p) is an interval with at least 3 edges, hence the ring FH (and then FG) is not serial. It remains to consider the case p = 2, 3. If p = 2, then G = GL(2, 2) ∼= S3 is 2-nilpotent, hence the ring FG is serial. Similarly for p = 3 the group GL(2, 3) has order 48 and is 3-solvable, hence FG is serial. Thus we may assume that p does not divide q. Let d be the order of q modulo p, i.e. the least d such that p | qd − 1. By the assumption we have 1 6 d 6 n, and clearly d | p− 1. We will show that d cannot be very small (otherwise the p-Sylow subgroup P of G is not cyclic) and cannot be very large (otherwise the Brauer tree of the principal block has too many edges). The description of normalizers and centralizers of p-Sylow subgroups of GL(n, q) is well known (see [21,22]). We will add some explanations to ease reader’s task. Lemma 2. 1) P is cyclic if and only if n < 2d. 2) If n < 2d then the factor group NG(P )/CG(P ) has order d. Proof. Consider the Galois field Fqd as a vector space (of dimension d) over Fq with a basis v1, . . . , vd. Let z be nonzero element of Fqd . Then zvi = ∑ j zijvj for some zij ∈ Fq. The mapping z 7→ (zij) defines an embedding of the multiplicative group of Fqd into GL(d, q). The image of a generator of F∗ qd gives us a matrix x ∈ GL(d, q) of order qd − 1. Write qd − 1 = pa · s such that p and s are coprime, hence y = xs generates the p-Sylow subgroup P of order pa. 1) If n > 2d, then one could insert in GL(n, q) two copies of GL(d, q) as 1 through d, and d + 1 through 2d diagonal blocks. It follows easily that P is not cyclic. 120 Serial group rings of finite groups On the other hand, if n < 2d then, comparing the sizes, we see that P can be chosen inside GL(d, q) embedded in the upper left 1 through d corner of GL(n, q), and therefore is generated by y. 2) It is known (see [22]) that the centralizer of P is generated by x, hence has order qd − 1. Furthermore (see [21, Lemma 4.6]) the normalizer of P is generated over CG(P ) by an element of order d. For our purposes it suffices to find an element which normalizes P and has order d modulo the centralizer. This can be achieved as follows. Suppose that the action of x on the basis is given by a matrix A = (aij), aij ∈ Fq: xvi = ∑ j aijvj . Applying the Frobenius morphism x 7→ xq on Fqd we obtain xqvq i = ∑ j aijv q j . It follows that the action of xq in the basis vq i is given by the same matrix A. Because in the original basis this action is given by Aq, we conclude that UAU−1 = Aq, where U is the transition (from vq i to vi) matrix. Then the conjugation by U defines an automorphism of order d on the subgroup generated by x. It follows that this action induces on P an automorphism ψ of the same order. Namely, let ψ(y) = yq and suppose that yqk = y for some k. Plugging y = xs we obtain x(qk−1)s = 1, therefore qd −1 = pa ·s divides (qk −1)s. It follows that pa divides qk −1, and hence d divides k, by the choice of d. Now we complete the proof of Theorem 1 by showing the following. Proposition 1. Let G = GL(n, q) and F is a field of characteristic p dividing the order of G but not dividing q. Then the group ring FG is serial if and only if n = 2, 3, p = 3 and q ≡ 2, 5 (mod 9). Proof. We may assume that the p-Sylow subgroup P of G is cyclic. By the item 1) of Lemma 2 it follows that n/2 < d 6 n, where d is the order of q modulo p. Suppose first that d > 2. If p = 2 it follows (since p does not divide q) that q is odd, therefore p divides q − 1 and d = 1, a contradiction. Thus we may assume that p > 2. By Fact 3 and the item 2) of Lemma 2 the Brauer tree of the principal block B0 of G has e = d edges. Furthermore [8, Prop. 4] implies that this tree is an interval. Since d > 2, this block is not serial. Thus we are left with the case d = 2, in particular p divides q2 −1. The definition of d yields that p does not divide q − 1 and hence divides q + 1. Again the Brauer tree of the principal block of G is an interval with e = 2 edges, whose exceptional vertex is located at its end. By Fact 3 the multiplicity of this vertex is (|P | − 1)/2. If |P | > 3 this bock is not serial. A. Kukharev, G. Puninski 121 Thus we may assume that |P | = 3, which clearly yields p = 3 and q ≡ 2, 5 (mod 9) (otherwise 9 divides the order of P ). It follows that the principal block is serial. We prove that, in this case, any non-principal block B of G is also serial. Namely, by Fact 3 the number of edges, e, of this block divides p − 1 = 2. If e = 1 then this block contains only one Brauer character, hence serial. If e = 2 then the multiplicity of the exceptional vertex equals (3 − 1)/2 = 1, hence this block is also serial. Note that in the proof of the implication ⇒ in Theorem 1 we used only that the principal block B0 of GL(n, q) is serial. 3. Special linear and projective special linear groups In this section we will consider the seriality of group rings of special linear groups SL(n, q) and projective special linear groups PSL(n, q). The answer turns out to be the same for both series; and the proofs go in parallel. Recall that SL(n, q) is a normal subgroup of GL(n, q) of index q − 1. Furthermore PSL(n, q) is obtained from SL(n, q) by factoring out the center Z whose order equals (n, q− 1). Note also that, except of PSL(2, 2) and PSL(2, 3), PSL(n, q) is a simple group. To avoid long sentences we will divide the classification theorem in two cases: when p divides q − 1 and when it is not. In the former case the answer is the same as in Theorem 1. Proposition 2. Let G is one of the groups SL(n, q) or PSL(n, q), n > 2. Let F be a field of characteristic p such that p does not divide q− 1. Then the ring FG is serial if and only if one of the following holds. 1) n = 2 and p = q equal 2 or 3. 2) n = 2, 3, p = 3 and q ≡ 2, 5 (mod 9). Proof. Since p does not divide q − 1, by Fact 2, we conclude that the seriality of group rings of SL(n, q) and GL(n, q) is equivalent. Applying Theorem 1 we obtain the desired conclusion for SL(n, q). Thus we may assume that G = PSL(n, q). If 1) or 2) holds true then the group ring R of SL(n, q) is serial. Since G is a factor group of this group, it follows that the group ring of G is a factor ring of R, therefore is also serial. Thus we may assume that the group ring of PSL(n, q) is serial and we need to show that either 1) or 2) holds true. 122 Serial group rings of finite groups By Fact 4 the principal block b0 of SL(n, q) has Z in its kernel, and therefore coincides with the principal block of PSL(n, q). Furthermore, because SL(n, q) contains the p-Sylow subgroup of GL(n, q) it follows by Fact 2 that the principal block B0 of GL(n, q) is serial. Now the result follows from the proof of Theorem 1 (see a remark at the end of Section 2). Now we consider the remaining case p | q − 1. In this case serial rings occur more often than in the GL-case (cp. Theorem 1). Proposition 3. Let G be one of the group SL(n, q) or PSL(n, q), n > 2 and let F be a field of characteristic p dividing q − 1. The group ring FG is serial if and only if n = 2 and p 6= 2. Proof. If n > 3 then it is easily seen that p-Sylow subgroups of G are not cyclic. Thus we may assume that n = 2. If G = PSL(2, q) then FG is serial if and only if p 6= 2 [12]. Thus we may assume that G = SL(2, q). If p = 2 then the group ring FG is not serial. Indeed, otherwise, being a factor ring of FG, the group ring of PSL(2, q) would be serial, a contradiction. It remains to consider the case p > 2 and we have to prove that the group ring of FG is serial. Observe that, if q is even, then SL(2, q) ∼= PSL(2, q), hence the ring is serial. Thus we assume that q is odd. In this case the center Z of SL(2, q) consists of matrices ±I, where I = ( 1 0 0 1 ). For the remaining part of the proof we need the character table of G = SL(2, q) — see Table 1. In the table, 1 6 l 6 (q − 3)/2, 1 6 m 6 (q − 1)/2, ε = (−1)(q−1)/2, ρ is a primitive (q− 1)-th root of 1, and σ is a primitive (q+ 1)-th root of 1. Let ν be a generator of the group F ∗ q . Denote γ = ( 1 0 1 1 ), δ = ( 1 0 ν 1 ), α = ( ν 0 0 ν−1 ) . So, the order of α is q − 1. The group G contains also an element β of order q + 1. Moreover, two columns for the classes of γ′ = −I · γ and δ′ = −I · δ are omitted (to save space in the table). The values of any irreducible character χ of G on these classes are obtained by the formulas χ(γ′) = χ(γ)χ(−I)/χ(I) and χ(δ′) = χ(δ)χ(−I)/χ(I). Since p | q − 1, only the sixth column of the table contain p-singular elements. In particular, the cyclic group 〈α〉 contains a generator y of a p-Sylow subgroup P of G. It is easy to show (see [5, p. 230]) that CG(y) = 〈α〉 and NG(y) = 〈α, ( 0 1 −1 0 ) 〉. Hence |NG(P )/CG(P )| = 2 . In particular, the number of A. Kukharev, G. Puninski 123 Classes I −I γ δ αl βm Number of classes 1 1 1 1 q−3 2 q−1 2 Size of classes 1 1 q2−1 2 q2−1 2 q(q+1) q(q − 1) 1G 1 1 1 1 1 1 ψ q q 0 0 1 −1 χi (i = 1, . . . , q−3 2 ) q+ 1 (−1)i × ×(q+1) 1 1 ρil + ρ−il 0 θj(j = 1, . . . , q−1 2 ) q− 1 (−1)j × ×(q−1) −1 −1 0 −(σjm + σjm) ξ1, ξ2 q+1 2 ε(q+1) 2 1±√ εq 2 1∓√ εq 2 (−1)l 0 η1, η2 q−1 2 − ε(q−1) 2 −1±√ εq 2 −1∓√ εq 2 0 (−1)m+1 Table 1. The character table of SL(2, q), q is odd [5, p. 228] edges in the principal block B0 of G equals 2, furthermore the number of edges in any block of G divides 2. Observe that θj , η1 and η2 have value 0 on the class of α. By [17, Theorem 4.4.14], these characters belong to blocks of defect zero. It follows that these blocks contain only one irreducible ordinary character, hence is serial. Furthermore it is easily checked (using [11, Theorem 2.1.8]) that the Steinberg character ψ belongs to the principal block B0. Looking at the values on p-singular elements (and using cross-naught business — see [11, Chap. 2]) we see that the Brauer tree of B0 is an interval with 2 edges having 1G and ψ at its ends. Thus if there is an exceptional vertex it should be located at the center of this interval (in fact certain characters χi will occupy the center making an exceptional vertex there). Because each character χi has the largest possible degree, it follows from [11, Lemma 2.1.22] that such a character cannot occur at the end of an interval of length 2. Thus the only possibility for such an interval is to have ξ1 at one end, ξ2 at another end, and some characters χi in between. But this block is clearly serial. In fact such a block exists if q ≡ 1 (mod 4); otherwise each non- principal block contains at most one modular character (i.e. its Brauer tree has at most one edge). By this we have established that the group ring of SL(2, q) is serial if 2 6= p | q − 1, hence finished the proof of the proposition. 124 Serial group rings of finite groups Prepositions 2 and 3 completely describe groups of SL(n, q) and PSL(n, q)-series whose p-modular group rings are serial. References [1] J.L. Alperin, Local Representation Theory, Cambridge University Press, 1981. [2] F.W. Anderson, K.R. Fuller, Rings and Categories of Modules, 2d edition, Springer Graduate Texts in Math., Vol. 13, 1992. [3] M. Auslander, I. Reiten, S. Smalø, Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, Vol. 36, Cambridge, 1995. [4] Y. Baba, K. Oshiro, Classical Artinian Rings, World Scient. Publ., 2009. [5] L. Dornhoff, Group representation theory. Part A: Ordinary Representation Theory (Pure and applied mathematics 7), New York, 1971. [6] D. Eisenbud, P. Griffith, Serial rings, J. Algebra, 17 (1971), pp. 389–400. [7] W. Feit, The Representation Theory of Finite Groups, North Holland Mathematical Library, Vol. 25, 1982. [8] P. Fong, B. Srinivasan, Blocks with cyclic defect groups in GL(n, q), Bull. Amer. Math. Soc., 3 (1980), pp. 1041–1044. [9] P. Fong, B. Srinivasan, Brauer trees in GL(n, q), Math. Z., 187 (1984), pp. 81–88. [10] D.G. Higman, Indecomposable representations at characteristic p, Duke J. Math., 21 (1954), pp. 377–381. [11] G. Hiss, K. Lux, Brauer Trees of Sporadic Groups, Oxford, 1989. [12] A. Kukharev, Seriality of group rings of unimodular projective groups, in: Proc. of the 71th Scientific Conf. of Students and PhD students of Belarusian State University, Minsk, May 18–21, 2014, Part 1, pp. 11–14. [13] A. Kukharev, G. Puninski, Serial group rings of finite groups. p-solvability, Algebra Discrete Math., 16 (2013), pp. 201–216. [14] A. Kukharev, G. Puninski, The seriality of group rings of alternating and symmetric groups, Vestnik of Belarusian State University, Mathemathics and Informatics series, 2 (2014), pp. 61–64. [15] A. Kukharev, G. Puninski, Serial group rings of finite groups. Sporadic simple groups and Suzuki groups, Notes Research Semin. Steklov Institute Sanct-Petersb., 435 (2015), pp. 73–94. [16] A. Kukharev, G. Puninski, Yu. Volkov, The seriality of the group ring of a finite group depends only of characteristic of the field, Notes Research Semin. Steklov Institute Sanct-Petersb., 423 (2014), pp. 57–66. [17] K. Lux, H. Pahlings, Representations of Groups: a Computational Approach, Cambrodge Studies in dvanced Mathmeatics, Vol. 124, 2010. [18] K. Morita, On group rings over a modular field which possess radicals expressible as principal ideal, Sci. Repts. Tokyo Daigaku, 4 (1951), pp. 177–194. [19] G. Puninski, Serial Rings, Kluwer, 2001. [20] B. Srinivasan, On the indecomposable representations of a certain class of groups, Proc. Lond. Math. Soc., 10 (1960), pp. 497–513. A. Kukharev, G. Puninski 125 [21] M. Stather, Constructive Sylow theorems for the classical groups, J. Algebra, 316 (2007), pp. 536–559. [22] A.J. Weir, Sylow p-subgroups of the classical groups over finite fields with chacter- istic prime to p, Proc. Amer. Math. Soc. 6 (1955), pp. 529–533. Contact information A. Kukharev, G. Puninski Faculty of Mechanics and Mathematics, Belarusian State University, 4, Nezavisimosti Ave., Minsk, 220030, Belarus E-Mail(s): kukharev.av@mail.ru, punins@mail.ru Web-page(s): www.mmf.bsu.by Received by the editors: 11.05.2015 and in final form 12.07.2015.

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