We call G to be an α,-Coset group, if it contains a proper α,-invariant normal subgroup N such that Nxα,= {xg | g ∈,G}, for some automorphism α,of G and any x ∈,G\N. Clearly, if α,is identity automorphism of G, one obtains the notion of con-cos groups, which was first introduced by Muktibodh in 2006. In the present article, we discuss some properties of the new notion. Also, we introduce the concept of α,-Camina groups and give its connection with the groups of property P, where P is the class of all finite groups such that their α,-centres are the same as α,-commutator subgroups of order p.

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We study the double Cosets of a Lie group by a compact Lie subgroup. We show that a Weil formula holds for double Coset Lie hypergroups and show that certain representations of the Lie group lift to representations of the double Coset Lie hypergroup. We characterize smooth (analytic) vectors of these lifted representations.

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We will introduce and study several algebraic properties of the covariance Cosets in C*-algebras. Indeed, we will characterize the covariance Coset in terms of commutators. Also, we will show that for an invertible element b, the covariance Coset of b-1 coincides with the covariance Coset of b*. Moreover, if b is normal, then the covariance Coset of b coincides with the covariance Coset of b*. In addition, we will prove that the covariance Coset is a cone.

There is much interest in finding short presentations for the finite simple groups. Indeed it has been suggested that all these groups are efficient in a technical sense. In previous papers we produced nice efficient presentations for all except one of the simple groups with order less than one million. Here we show that all simple groups with order between 1 million and 5 million are efficient by giving efficient presentations for all of them. Apart from some linear groups these results are all new. We also show that some covering groups and some larger simple groups are efficient. We make substantial use of systems for computational group theory and, in particular, of computer implementations of Coset enumeration to find and verify our presentations.

This paper introduces some new rank six geometries associated with the Fischer sporadic simple group Fi22. These geometries are both residually connected and firm, with Fi22 acts as a flag-transitive automorphism group on them. The previously known geometries for Fischer sporadic simple group Fi22 have rank at most four. Therefore, we investigate improvements to the lower bound of the maximum rank of the residually connected and firm geometries on which the group Fi22 acts as a flag-transitive automorphism group. Moreover, we demonstrate that the independent generating set of Fischer sporadic simple group Fi22 has a size of at least seven.

In this article, we study connections between components of the Cayley graph Cay(G, A), where A is an arbitrary subset of a group G, and Cosets of the subgroup of G generated by A. In particular, we show how to construct generating sets of G if Cay(G, A) has finitely many components. Furthermore, we provide an algorithm for finitely minimal generating sets of finite groups using their Cayley graphs.