Finite groups with small unbalancing 2-components

Gilman, Robert H.; Solomon, Ronald

doi:10.2140/pjm.1979.83.55

Cited by 4 publications

(3 citation statements)

References 16 publications

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“…In [1348] with the help of solution of the p.s.f, for A n he obtained an additional reduction of the U-conjecture limiting possible types of L/O(L) to the following list: L2(q), q odd, A 7, He, and the covering group for L3 (4). In this restricted list Gilman and Solomon [761,764] introduced the concept and proved the existence in G of a maximal unbalanced triple. The proof of the U-conjecture proceeded further along the path of passing from a maximal unbalanced triple to a maximal 2-component (the centralizer of some involution) in G, the concept of which was first introduced in [1346].…”

Section: B Classification Of Simple Groupsmentioning

confidence: 99%

“…The proof of the U-conjecture proceeded further along the path of passing from a maximal unbalanced triple to a maximal 2-component (the centralizer of some involution) in G, the concept of which was first introduced in [1346]. In [764] modulo solution of the p.s.f, of type L3(4) and L2(q) in G a maximal 2-component M was constructed such that either M/O(M) = L2(q), A 7 or He, or M/Z*(M) = L~(4). In the case M/O(M) = L2(q) or A 7 Foote [726] showed that the 2-rank of C6[M/O(M)] is equal to I, and then in the works of Harris and Solomon [856,867] and Fritz [740] it was shown, in particular, that M/O(M) ~ A 7 and…”

Section: B Classification Of Simple Groupsmentioning

confidence: 99%

“…Griess and Solomon [821] completed the reduction of the U-conjecture to the case M/O(M) L2(q) and the solution of the p.s.f, for components of the type La(4) and L2(q). Finally, in the works of Harris [861] and Gilman and Solomon [764] a final reduction of the U-conjecture was obtained to the solution of the p.s.f, for the components of the type L3(4), L~(3), U4(3), P~7(3), P~8• Another approach to the proof of the B-conjecture is contained in the work of Walter [1486, there is a preprint]. The differences between the conceptually closely related approaches of Thompson and Walter consist in the following.…”

Section: B Classification Of Simple Groupsmentioning

confidence: 99%

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Finite groups

Кондратьев¹,

Махнев²,

Starostin³

1989

J Math Sci

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The present survey is formed primarily on the basis of works reviewed in the Referativnyi Zhurnal "Matematika" during 1976-1983 and is a continuation of surveys of the same name published in 1966, 1971, 1976 in the series "Albegra, Topologiya, Geometriya" (INT). Principal attention is devoted to finite simple groups and their classification.The present survey is formed primarily on the basis of works reviewed in RZh "Matematika" during 1976-1983 and is a continuation of surveys of the same title of A. I. Kostrikin [162], So A. Chunikhin, L. A. Shemetkov [371], and V. D. Mazurov [194]. Rapid development of the theory of finite groups was observed during this period. More than 2000 papers were published and also the books [

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Section: B Classification Of Simple Groupsmentioning

confidence: 99%

Section: B Classification Of Simple Groupsmentioning

confidence: 99%

Section: B Classification Of Simple Groupsmentioning

confidence: 99%

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Finite groups

Кондратьев¹,

Махнев²,

Starostin³

1989

J Math Sci

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Standard components of type F3

Syskin¹

1981

Algebra and Logic

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Finite groups with unbalancing 2-components of {L̂3(4), He}-type

Griess¹,

Solomon²

1979

Journal of Algebra

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Considerable progress has been made in recent years towards the classification of finite simple grnups of odd Chevallcy type These groups may be defined as simple groups G containing an involution t such that C,(t)jO(C,(t)) has a subnormal quasi-simple subgroup. This property is possessed by most Chevalley groups over fields of odd order but by no simple Chevalley groups over fields of characteristic 2, hence the name. It is also a property of alternating groups of degree at least 9 and eighteen of the known sporadic simple groups. Particular attention in this area has focussed on the so-called B-Conjecture. R-Conjecture: Let G be a finite group with O(G) = (1 j. Let t be an involution of G andL a pcrfcct subnormal subgroup of CJt) withL/O(fJ) quasi-simple, then L is quasi-simple. The R-Conjecture would follow as an easy corollary of the Unbalanced Group Conjecture (.5'-Conjecture). U-Chjecture: Let G be a finite group withF*(G) quasi-simple. Suppose there is an involution t of G with O(C,(t)) e O(G). Then F*(G)/F(G) is isomorphic to one of the following: (I) A Chevalley group of odd characteristic. (2) ,4n alternating group of odd degree. (3) T,,(4) nr Held's simple group, He. LVith the exception of the,!,,(q) component case, the cases (1) and (2) have been complctcly handled in [3], [5], [7], [8], [20], [29] and [30]. The L,(q) problem has been reduced in [19] to the solution of a small number of specific standard form problems in groups satisfying the I/'-Conjecture. This paper treats case (3). Hcfnrc we state orrr main results, we need to make some definitions. A perfect x Partially supported by KSI: Grant MCS76-07280. t Partially supported by .XSk' &ant ~&lCS75-08346.

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Finite groups with small unbalancing 2-components

Cited by 4 publications

References 16 publications

Finite groups

Finite groups

Standard components of type F3

Finite groups with unbalancing 2-components of {L̂3(4), He}-type

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