The purpose of this note is to determine all endotrivial modules in prime characteristic p, for a finite group having a cyclic Sylow p-subgroup. In other words, we describe completely the group of endotrivial modules in that case. The endotrivial modules are a family of finitely generated modules for a group algebra of a finite group which appear naturally in modular representation theory. Seve...

Left braces, introduced by Rump, have turned out to provide an important tool in the study of set-theoretic solutions quantum Yang–Baxter equation. In particular, they allowed construct several new families solutions. A left brace ( B , + ? ) is a structure determined two group structures on set : abelian and satisfying certain compatibility conditions. The main result this paper shows that eve...

It is common in mathematics for a subject to have its local and global aspects; such is the case in group theory. For example, the structure and embedding of subgroups of a group G may be usefully thought of as part of the local structure of G while the normal subgroups, quotient groups and conjugacy classes are relevant to the global structure of G. Furthermore, the connections between local a...

We prove that a finite group G $G$ has normal Sylow p $p$ -subgroup P $P$ if, and only every irreducible character of appearing in the permutation ( 1 ) $({\bf 1}_P)^G$ with multiplicity coprime to degree . This confirms prediction by Malle Navarro from 2012. Our proof above result depends on reduction simple groups ultimately combinatorial analysis properties branching coefficients for symmetr...

Abstract A finite group G G is called an MSN-group if all maximal subgroups of the Sylow are subnormal in . In this article, we investigate structure groups such that a non-MSN-group even order which every subgroup MSN-group. addition, determine minimal simple whose second MSN-groups.

On the other hand, for control of fusion and other locally controlled phenomona, Alperin-Goldschmidt subpairs play a crucial role. We take this opportunity to remind the reader that when B is a p-block, an Alperin-Goldschmidt B-subpair (U, bU ) is a B-subpair such that bU has defect group Z(U) and Op(NG(U, bU )/UCG(U)) = 1. When B is the principal p-block, the AlperinGoldschmidt B-subpairs may ...

A conjecture of Michel Brou e states that if D is an abelian Sylow p-subgroup of a nite group G, and H = N G (D), then the principal blocks of G and H are Rickard equivalent. The structure of groups with abelian Sylow p-subgroups, as determined by P. Fong and M.E. Harris, raises the following question: assuming that Brou e's conjecture holds for the simple components of G, under what conditions...

This paper aims to study the Smarandache cosets and derive some interesting results about them. We prove the classical Lagranges theorem for Smarandache semigroup is not true and that there does not exist a one-to-one correspondence between any two right cosets. We also show that the classical theorems cannot be extended to all Smarandache semigroups. This leads to the definition of Smarandache...

We consider 2-local geometries and other subgroup complexes for sporadic simple groups. For six groups, the fixed point set of a noncentral involution is shown to be equivariantly homotopy equivalent to a standard geometry for the component of the centralizer. For odd primes, fixed point sets are computed for sporadic groups having an extraspecial Sylow p-subgroup of order p, acting on the comp...