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Finite groups with a quasisimple component of type $PSU(3,2^{n})$ on elementary abelian form

1975; Duke University Press; Volume: 19; Issue: 2 Linguagem: Inglês

10.1215/ijm/1256050811

1945-6581

Peter Landrock,

Advanced Algebra and Geometry

It is a quite common phenomenon among sporadic simple groups that some involution has a centralizer with a quasisimple component of even characteristic which is on elementary abelian form.By this we mean that the centralizer of the component has an elementary abelian Sylow 2-subgroup.(For definition of component, quasisimple etc., we refer the reader to for example D. Gorenstein's survey article on finite simple groups.)Examples of such sporadic simple groups are: Janko's first group Jx (Z2 x PSL(2, 4)), the Mathieu group Mx2 (2 2 X $5) the Hall-Janko group J2 (Z2 X Z 2 x PSL(2, 4)), the sporadic Suzuki group Su (Z2 x Z2 x PSL(3, 4)), Held's group He (a central extension of PSL(3, 4) by Z2 x Z2), Rudvalis' group Ru (Z2 x Z2 x Sz(8)), Conway's group Cox (Z2 x Z2 x G2(4)) and Fischer's new simple group F2(? (Z2 x Z2 x F4(2)).This gives rise to several classification problems, among which is the following natural one.Classify finite (in particular simple) groups with an involution whose central- izer C is isomorphic to the direct product of an elementary abelian 2-group E and a group B containing a normal subgroup Bo which is quasisimple of Bender-type such that CB(Bo) Z(Bo).However, to deal with this problem we need an additional assumption on the involutions of E. A natural one, at least when B o is of Bender-type, seems to be that C is the centralizer of all the involutions in E (trivially satisfied when IEI 2.) This is a type of problem which for instance occurs in a recent work by D. Mason, in which he considers finite simple groups all of whose components are of Bender-type (and the centralizer of some involution not 2-constrained of course).Furthermore, J2 and Ru satisfy this assumption.Exactly this problem has been considered in the following cases when B 0 is isomorphic to one of the simple groups PSL(2, q) or Sz(q), B Bo and G is simple: E -Z 2 and B -PSL(2, 2"), by Z. Janko, B _ PSL(2, 2n), by F. L. Smith, B -Sz(q), by U. Dempwolff, and some as special cases in related problems which have been dealt with by M. Aschbacher and K. Harada.Here we shall answer the question completely for all groups with B0 quasi- simple of PSU(3, 2")-type, the third class of groups of Bender-type.