For a maximal subgroup M of a finite group G , the normal index r¡(G : M) is defined to be the order of a chief factor H/K where H is minimal in the set of supplements of M in G . We obtain several results on the normal index of maximal subgroups M of composite index in G with [G : M]p = 1 which imply G to be solvable, supersolvable.

we call $h$ an $ss$-embedded subgroup of $g$ if there exists a‎ ‎normal subgroup $t$ of $g$ such that $ht$ is subnormal in $g$ and‎ ‎$hcap tleq h_{sg}$‎, ‎where $h_{sg}$ is the maximal $s$-permutable‎ ‎subgroup of $g$ contained in $h$‎. ‎in this paper‎, ‎we investigate the‎ ‎influence of some $ss$-embedded subgroups on the structure of a‎ ‎finite group $g$‎. ‎some new results were obtained.‎

This paper investigates the asymptotic behaviour of the minimal number of generators of finite index subgroups in residually finite groups. We analyze three natural classes of groups: amenable groups, groups possessing an infinite soluble normal subgroup and virtually free groups. As a tool for the amenable case we generalize Lackenby’s trichotomy theorem on finitely presented groups.

Let G be a finite group with a non-abelian minimal normal subgroup N which is a direct product of the simple group X. The maximal subgroups of G which complement N and their conjugacy classes are parametrised in terms of certain homomorphisms taking values in AutX and satisfying particular conditions.

An elusive permutation group is a transitive permutation group with no fixed point free elements of prime orders. In this paper we proved that each minimal normal subgroup of an elusive permutation group with squarefree degree is abelian. As an almost immediate consequence, it follows that there are no elusive permutation groups of degree 2pq where p, q are distinct odd primes.

Let p be a prime. A group is called p-closed if it has a normal Sylow p-subgroup and it is called p-exponent closed if the elements of order dividing p form a subgroup. A group is minimal non-p-closed if it is not p-closed but its proper subgroups and homomorphic images are. Similarly, a group is called minimal non-p-exponent closed if it is not p-exponent closed but all its proper subgroups an...

Sensory Guillain-Barré syndrome (GBS) is a rare heterogeneous subgroup of GBS characterized by the primary involvement sensory neurons resulting in distinctive clinical presentation. usually occurs with acute and monophasic symptoms, no or minimal muscle weakness. patient show hypo- areflexia, distal paresthesia, normal cerebrospinal fluid finding albumino-cytologic dissociation which are sugge...