We consider a ramified Galois cover φ : X̂ → Px of the Riemann sphere Px, with monodromy group G. The monodromy group over Px of the maximal unramified abelian exponent n cover of X̂ is an extension nG̃ of G by the group (Z/nZ), where g is the genus of X̂. Denote the set of linear equivalence classes of divisors of degree k on X̂ by Pic(X̂) = Pic. This is equipped with a natural G action. We show tha...

We prove that the number of groups of order p whose Frattini subgroup is central is for fixed n a PORC (‘polynomial on residue classes’) function of p. This extends a result of G. Higman.

‎‎in this paper we determine all finite $2$-groups of‎ ‎class $2$ in which every automorphism of order $2$ leaving the frattini subgroup elementwise fixed is inner‎.

In group theory nilpotency of a group has a great importance. In this paper we have studied some concept of nilpotency through right transversals. We have also studied prime power groups and frattini subgroups through right transversals.