It is shown that the coset lattice of a finite group has shellable order complex if and only if the group is complemented. Furthermore, the coset lattice is shown to have a Cohen–Macaulay order complex in exactly the same conditions. The group theoretical tools used are relatively elementary, and avoid the classification of finite simple groups and of minimal finite simple groups. © 2006 Elsevi...

the non-commuting graph $nabla(g)$ of a non-abelian group $g$ is defined as follows: its vertex set is $g-z(g)$ and two distinct vertices $x$ and $y$ are joined by an edge if and only if the commutator of $x$ and $y$ is not the identity. in this paper we 'll prove that if $g$ is a finite group with $nabla(g)congnabla(bs_{n})$, then $g cong bs_{n}$, where $bs_{n}$ is the symmetric group of degre...

There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. The most successful approach to this conjecture has been Borovik’s program analyzing a minimal counterexample, or simple K-group. We show that a simple K-group of finite Morley rank and odd type is either aalgebraic of else has Prufer rank at mos...

A p-regular element in a finite group is an element of order not divisible by the prime number p. We show that for every prime p and every finite simple group S, a fair proportion of elements of S is p-regular. In particular, we show that the proportion of p-regular elements in a finite classical simple group (not necessarily of characteristic p) is greater than 1/(2n), where n− 1 is the dimens...

Let G = (V,E) be a simple graph with exactly n vertices and m edges. The aim of this paper is a new method for investigating nontriviality of the automorphism group of graphs. To do this, we prove that if |E| >=[(n - 1)2/2] then |Aut(G)|>1 and |Aut(G)| is even number.

Let G be a finite group and let d(G) be the minimal number of generators for G. It is well known that d(G) = 2 for all (non-abelian) finite simple groups. We prove that d(H) ≤ 4 for any maximal subgroup H of a finite simple group, and that this bound is best possible. We also investigate the random generation of maximal subgroups of simple and almost simple groups. By applying a recent theorem ...

‎Let $G $ be a finite group and $X$ be a conjugacy class of $G.$ The‎ ‎rank of $X$ in $G,$ denoted by $rank(G{:}X),$ is defined to‎ ‎be the minimal number of elements of $X$ generating $G.$ In this‎ ‎paper we establish the ranks of all the conjugacy classes of‎ ‎elements for simple alternating group $A_{10}$ using the structure‎ ‎constants method and other results established in‎ ‎[A.B.M‎. ‎Bas...

Attempt 3 questions. (If you attempt more, only the best 3 will be counted.) All questions carry the same number of marks. Unless otherwise stated, all groups are compact (or finite), and all representations are of finite degree over C. 1. Define a group representation. What is meant by saying that a representation α is simple? Determine all simple representations of S 3. from first principles....