For a finite group G and an inverse-closed generating set C of G, let Aut(G;C) consist those automorphisms which leave invariant. We define Aut(G;C)-invariant normal subgroup Φ(G;C) has the property that, for any generators if we remove from it all elements Φ(G;C), then remaining is still G. The contains Frattini Φ(G) but inclusion may be proper. Cayley graph Cay(G,C) edge-transitive acts trans...

Let G be a finite group and let K = x the conjugacy class of an element G. In this paper, it is proved that if N normal subgroup such coset union ? 1 (the inverse x), then ? ? are solvable. As application, we prove there exists natural number n ? 2 ? ,

By means of Dubois and Prade’s idea, we introduce the concept of rough fuzzy normal subgroups and investigate some related properties. Combining rough sets and soft sets, rough soft normal subgroups with respect to a normal subgroup of groups in Pawlak approximation spaces are introduced. Moreover, we investigate the relationships between lower and upper rough soft normal subgroups with respect...