INTERNATIONAL JOURNAL OF GROUP THEORY
Year:2012 | Volume:1 | Issue:1|Papers Count:7

Let G be a finite group. We say that G has spread r if for any set of distinct non-trivial elements of G X: = {x1; . . . ; xr} Ì G# there exists an element y ϵ G with the property that áxi, yñ=G for every 1 £ i £ r. We say G has exact spread r if G has spread r but not r+1. The spreads of finite simple groups and their decorations have been much-studied since the concept was first introduced by Brenner and Wiegold in the mid 1970s. Despite this, the exact spread of very few finite groups, and in particular of the finite simple groups and their decorations, is known. Here we calculate the exact spread of the sporadic simple Mathieu group M23, proving that it is equal to 8064. The precise value of the exact spread of a sporadic simple group is known in only one other case - the Mathieu group M11.

In the 1960's Noboru Iwahori and Hideya Matsumoto, Eiichi Abe and Kazuo Suzuki, and Michael Stein discovered that Chevalley groups G = G (F, R) over a semilocal ring admit remarkable Gauss decomposition G = TUU-U, where T = T (F, R) is a split maximal torus, whereas U = U (F, R) and U- = U- (F, R) are unipotent radicals of two opposite Borel subgroups B = B (F, R) and B- = B- (F, R) containing T. It follows from the classical work of Hyman Bass and Michael Stein that for classical groups Gauss decomposition holds under weaker assumptions such as sr (R) = 1 or asr (R) = 1. Later the third author noticed that condition sr (R) =1 is necessary for Gauss decomposition. Here, we show that a slight variation of Tavgen's rank reduction theorem implies that for the elementary group E = E (F, R) condition sr (R) =1 is also sufficient for Gauss decomposition. In other words, E = HUU-U, where H = H (F, R) =T Ç E. This surprising result shows that stronger conditions on the ground ring, such as being semi-local, asr (R) = 1, sr (R, L) = 1, etc., were only needed to guarantee that for simply connected groups G = E, rather than to verify the Gauss decomposition itself.

Let V be a vector space over a field F of characteristic p ³ 0 and let T be a regular subgroup of the affine group AGL (V). In the finite dimensional case we show that, if T is abelian or p > 0, then T is unipotent. For T abelian, pushing forward some ideas used in [A. Caranti, F. Dalla Volta and M. Sala, Abelian regular subgroups of the affine group and radical rings, Publ. Math. Debrecen 69 (2006), 297-308.], we show that the set {t - I | t Î T} is a subalgebra of EndF (F Å V), which is nilpotent when V has finite dimension. This allows a rather systematic construction of abelian regular subgroups.

Given a non-abelian finite group G, let p (G) denote the set of prime divisors of the order of G and denote by Z (G) the center of G. The prime graph of G is the graph with vertex set p (G) where two distinct primes p and q are joined by an edge if and only if G contains an element of order pq and the non-commuting graph of G is the graph with the vertex set G-Z (G) where two non-central elements x and y are joined by an edge if and only if xy ¹ yx. Let G and H be non-abelian finite groups with isomorphic non-commuting graphs. In this article, we show that if |Z (G)| = |Z (H)|, then G and H have the same prime graphs and also, the set of orders of the maximal abelian subgroups of G and H are the same.

It is shown that there are exactly seventy-eight 3-generator 2-groups of order 211 with trivial Schur multiplier. We then give 3-generator, 3-relation presentations for forty-eight of them proving that these groups have deficiency zero.

We pursue further our investigation, begun in [H. Smith, Groups with all subgroups subnormal or nilpotent-by-Chernikov, Rend. Sem. Mat. Univ. Padova 126 (2011), 245-253] and continued in [G. Cutolo and H. Smith, Locally finite groups with all subgroups subnormal or nilpotent-by-Chernikov. Centr. Eur. J. Math. (to appear)] of groups G in which all subgroups are either subnormal or nilpotent-by-Chernikov. Denoting by X the class of all such groups, our concern here is with locally finite p-groups in the class X, where p is a prime, while an earlier article provided a reasonable classification of locally finite X-groups in which all of the p-sections are nilpotent-by-Chernikov. Our main result is that if G is a Baer p-group in X then G is nilpotent-by-Chernikov.

In this paper we study right n-Engel group elements. By modifying a group constructed by Newman and Nickel, we construct, for each integern n ³ 5, a 2-generator group G=áa, bñ with the property that b is a right n-Engel element but where [bk,n a] is of infinite order when k Ï {0, 1}.