Powers and Skew Braces for Classical Groups

Abstract

This thesis is divided into two parts. The first part concerns generating functions for M-powers (M \geq 2) in finite symplectic and orthogonal group. We will be giving generating functions for the separable, semisimple, cyclic, and regular conjugacy classes (and hence elements) in the concerned group. This enables us to find the corresponding probability with the help of generating functions. The second part is concerned with skew braces corresponding to the groups of the form Zn \rtimes Z2. Fixing this group to be the additive group (resp. multiplicative group), we find the multiplicative group (resp. additive group), such that they form a skew brace when n is odd. A complete classification is obtained when we assume that the radical Ra(n) is a Burnside number.