Centralizer Of Involution Research Articles

A note on centralizers of involutions involving simple groups

If a finite group G has a ‘central’ involution t whose centralizer in G is (t) × H then, under certain conditions on H, G cannot be simple.

Finite simple groups with Sylow 2-subgroups of typePSL(5,q),qodd

The purpose of this paper is to study finite simple groups whose Sylow 2-subgroups are isomorphic to those ofPSL(5,q) for some oddq.This work originally appeared in the author's Ph.D. Thesis, (14), and at that time, the structure of the centralizers of involutions was worked out – see below for details. Subsequently, in view of the fact that ifq≡ – 1 (mod 4), the Sylow 2-subgroup ofPSL(5,q) has sectional 2-rank 4 (being, in fact, isomorphic to the wreath product of a semidihedral group by a cyclic group of order 2), the complete classification, at least for this case, was needed for Gorenstein and Harada's monumental work on sectional 2-rank at most 4, (8), and therefore Collins and Solomon completed the characterization ofPSL(5,q) andPSU(5,q) (for all oddq) in a neat paper, (5). Combining our result with theirs, we are able to state the following Theorem.

Groups with decomposable involution centralizers

Groups with decomposable involution centralizers

Centralizers involving Mathieu groups

A simple group G cannot contain a central involution t with CG(t) = 〈t〉 × M, where M is isomorphic to a simple Mathieu group.

Balance and generation in finite groups

In the study of finite simple groups, the analysis of the cores of the centralizers of involutions is of fundamental importance. The principal aim of this analysis is to prove that the centralizers of involutions have what is called semisimple 2-layers, a property which holds in all known simple groups. Thompson has termed this the B(G)-conjecture. At present two methods exist for treating this problem-the so-called signalixer functor method which has as its starting point the GorensteinGoldschmidt signalizer functor theorem [3-51 and the Bender method which emanates from his simplified proof of the uniqueness theorems of FeitThompson in minimal simple groups of odd order. Both methods have been used effectively in the applications, the former notably in the study of balanced groups [S] and in a number of characterization theorems concerning known simple groups and families of simple groups by the structure of their Sylow 2-subgroups; and the latter in Bender’s proof of the classification of simple groups with abelian Sylow 2-subgroups and Goldschmidt’s determination of all simple groups which possess a nontrivial abelian 2-subgroup which is strongly closed in a Sylow 2-subgroup (with respect to the given simple group). The prototype of a balanced group X is one in which the associated quasisimple components (if any) of the centralizers of the involutions of G are groups of Lie type over GF(2”) (with the exception of L,(4) and L,(4) and their covering groups, these being the only such groups which are not balanced quasisimple groups). In particular, X is balanced if the centralizer of every involution is 2-constrained (corresponding to the case in which no such components exist). The object of the present paper is to develop the

Finite Groups With Product Fusion

In several recent papers concerning the classification of finite simple groups, the investigators were forced to treat subsidiary problems involving a group G whose Sylow 2-subgroup S is a direct product S = S, x S2 in which the fusion of 2-elements of G corresponds to that of the direct product of two groups having S, and S2, respectively, as their Sylow 2-subgroups. For example, in the study of groups, with Sylow 2-subgroups of type G2(q) or Psp(4, q), q odd, the determination of the structure of the centralizer of a central involution requires a solution of this subsidiary problem in the case in which S, and S2 are dihedral groups ([9]). The analogous problem arises with S, dihedral and S2 wreathed in Mason's work on groups with Sylow 2-subgroups of type L4(q), q odd. Likewise in the analysis of groups with Sylow 2-subgroups of type Psp(6, q), q odd, two other such direct product problems arise in the course of determining the structure of the centralizers of involutions. It would therefore appear to be quite useful to have available an effective result about the structure of such a group G when the direct factors S, S2 of S are arbitrary. It is the object of this paper to establish such a general result. Suppose that we are trying to determine all simple groups G satisfying some set of conditions. In such a problem, it is very likely that one can reduce to the case in which the critical composition factors of the proper subgroups of G are of known type. Moreover, in those cases in which such a classification theorem has been obtained, specific properties of such composition factors have entered into the arguments. It is therefore reasonable and most likely necessary to impose some general conditions on appropriate composition factors of the proper subgroups of the group G under investigation. It turns out that a single assumption stated in terms of the notion of balance is all that is necessary. Moreover, we make this assumption only on the composition factors of certain subgroups of G which possess a Sylow 2-subgroup of the form T1 x T2with T, z Si, i = 1, 2.

A characterization of Ln(2 m) by the centralizer of one 2-central involution

A characterization of Ln(2 m) by the centralizer of one 2-central involution

A characterization of a finite simple Hall-Janko group

A finite simple Hall-Janko group in the class of simple groups is characterized by the structure of the centralizer of a central involution.

A characterization of odd-order extensions of the Steinberg groups D42( q3), q = 2 n, by the centralizer of one central involution

A characterization of odd-order extensions of the Steinberg groups D42( q3), q = 2 n, by the centralizer of one central involution

A CHARACTERIZATION OF M12 BY CENTRALIZER OF INVOLUTION

Journal Article A CHARACTERIZATION OF M12 BY CENTRALIZER OF INVOLUTION Get access G. N. THWAITES G. N. THWAITES 127 Dunchurch Road, Rugby Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 24, Issue 1, 1973, Pages 537–557, https://doi.org/10.1093/qmath/24.1.537 Published: 01 January 1973 Article history Published: 01 January 1973 Received: 05 February 1973

Groups with central 2-Sylow intersections of rank at most one

An involution in a finite group is called central if it lies in the center of a 2 2 -Sylow subgroup of G G . A 2 2 -Sylow intersection is called central if it is either trivial or contains a central involution. Suppose G G is a finite simple group all of whose central 2 2 -Sylow intersections are trivial or rank one 2 2 -groups. It is proved that G G is a known simple group.

Finite groups with 2-decomposable centralizers of involutions

Finite groups with 2-decomposable centralizers of involutions

A characterization of odd order extensions of G2(2 n) by the centralizer of one central involution

A characterization of odd order extensions of G2(2 n) by the centralizer of one central involution

Centralizers of involutions in balanced groups

Centralizers of involutions in balanced groups

FINITE GROUPS WITH 2-CLOSED OR 2'-CLOSED CENTRALIZERS OF INVOLUTIONS

Finite nonsolvable groups are described whose involution centralizers are 2-closed or 2′-closed, whereas the Sylow p-subgroups for p>2 are cyclic.

Simple finite groups of 2-rank 3 such that the centralizers of involutions are solvable

Simple finite groups of 2-rank 3 such that the centralizers of involutions are solvable

Finite groups with 2-closed or 2?-closed centralizers of involutions

Finite nonsolvable groups are described whose involution centralizers are 2-closed or 2′-closed, whereas the Sylow p-subgroups for p>2 are cyclic.

Finite simple groups with unique 2-length of centralizers of involutions

Finite simple groups with unique 2-length of centralizers of involutions

On finite groups with solvable centralizers of involutions

On finite groups with solvable centralizers of involutions

Some applications of the theory of blocks of characters of finite groups IV

Since it has now been proved by 11’. Feit and J. G. Thompson [IV] that finite groups of odd order are solvable, our attention will naturally turn to groups of finite even order. There are methods available here which do not have analogues in the case of an odd order. We may pose the following question: Given a Zgroup P, what can be said about the finite groups G which have P as their 2-Sylow group ? Usually, it will be assumed that also information concerning the centralizers of involutions is available. Still, it seems rather surprising that nontrivial results for G can be obtained in these circumstances. ‘There are diffcrcnt xvays in which the elements of 1’ can react to G and, because of this, it is natura1 to subdivide otlr problem. lcor instance, for each G, DC have the G-classes in P, i.c. the intersections of the conjugate classes of G with P. It is natural to assume that these G-classes of P are known. ‘J’herc are further conditions of this type, but it \vill not be necessary to go into this in this paper. After some preliminary discussions (Sections II-I\:), we discuss first two special cases. In Section VI, groups Pare considered which are direct products of two cyclic groups of order 2”’ ,:, 2. If the 2-Sylow group of P of G is a quaternion-group and if G has no normal subgroup of odd order greater than 1, it had been proved in Erauer-Suzuki 1181 that G has a center of order 2, see also Suzuki [21]. The proof was complicated in the case of an ordinary quaternion group and had only been sketched. In Section 1’11, a simplified proof is given. In the following sections, P may be an arbitrary 2-group. Choose arbitrarily an element v of P and two involutions (i.e., elements of order 2) y1 , yz of P,