SQUARING A CONJUGACY CLASS AND COSETS OF NORMAL SUBGROUPS.

Let G be a finite group and let K be the conjugacy class of x ∈ G. If K² is a conjugacy class of G, then [x, G] is solvable. If the order of x is a power of prime, then [x, G] has a normal p-complement. We also prove some related results on the solvability of certain normal subgroups when a non-trivial coset has certain properties. [ABSTRACT FROM AUTHOR]