Groups, transversals, and loops
A family of loops is studied, which arises with its binary operation in a natural way from some transversals possessing a ``normality condition''.
A family of loops is studied, which arises with its binary operation in a natural way from some transversals possessing a ``normality condition''.
This paper deals with a rationality condition for groups. Let n be a fixed positive integer. Suppose every element g of the finite solvable group is conjugate to its nth power g n. Let p be a prime divisor of the order of the group. We conclude that the multiplicative order of n modulo p is small, or p is small.