Two theorems on groups of characteristic 2-type (original) (raw)

1975, Pacific Journal of Mathematics

AI-generated Abstract

This paper addresses conjectures regarding finite simple groups that are of characteristic 2-type and characteristic 3-type. The main results extend previous theorems by providing proofs under the assumption of a minimum 2-local 3-rank. Specific conditions for constraints and properties of certain group structures are discussed, revealing new insights into the interplay of local and global group properties, particularly in relation to the structure of simple groups.

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3-local characterization of Held groups

Algebra and Logic, 1980

In the recent past a series of strong results have been announced, which essentially constitute an exhaustive treatment of the problem of describing the p -local structure of finite groups p of type characteristic two in the case where the G -rank ( p an odd prime), of the 2-local subgroups of G is sufficiently big (viz°, ~ ). The situation is much less clear in the case of small fl -rank. Here, it seems that a characterization would be useful of known simple groups, not necessarily of type characteristic two, by means of the centralizers of elements of order /D, or -in the first place -by means of the centralizers of elements of order three.

Notes on abelian groups. II

Acta Mathematica Academiae Scientiarum Hungaricae

w 5. p.basic subgroups of arbitrary abelian groups KULIKOV [8] introduced the notion of basic subgroups of abelian p-groups which has proved to be one of the most important notions in the theory of p-groups of arbitrary power. Basic subgroups can be defined in any module over the ring of p-adic integers, or, more generally, over any discrete valuation ring. Here we want to give a generalization of basic subgroups to any group so that it coincides with the old concept whenever the group is primary. In the general case, to every prime p, one can define p-basic subgroups where in the definition the prime p plays a distinguished role. The p-basic subgroups are not isomorphic for different primes, but are uniquely determined (up to isomorphism) by the group and the prime p. We shall see that p-basic subgroups are useful in certain investigations. Let G be an arbitrary (abelian) group l and p an arbitrary, but fixed prime. We call a subset [x~]~ea of G, not containing 0, p-independent, if for any finite subset xl .... ,x~ a relation nlxl-[-... q-nkx1~ EprG

Simple groups of order p · 3a · 2b

Journal of Algebra, 1970

In this paper some local group theoretic properties of a simple group G of order p * 3@ * 2b are found. These are applied in a later paper to show there are no simple groups of order 7 * 3" * 2b other than the three well-known ones. R. Brauer [4] has shown there are only the three known simple groups LI, , A, , and O,(3) of order 5 .3" * 2b. His treatment uses modular character theory especially for the prime 5. It follows from J. Thompson's X-group paper [9] that if G is a simple group of order p + 3" * 2*, then p = 5,7, 13, or 17. In our later treatment of the case p '=-T 7 we seem to need the results of the N-group paper itself. This present paper prepares the way.

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