On semi-rational Frobenius groups

Abstract

An element of a group G is called semi-rational if there is a natural number m such that each generator of 〈x〉 belongs to the conjugacy class of G containing x or xm. If all elements of G are semi-rational, then G is called a semi-rational group. In this paper, we study semi-rational Frobenius groups G and obtain results concerning effect of semi-rationality property on the kernel and complement of G. In particular, we show that ∣π(G)∣ ≤ 5 which answers Problem 2 in [D. Chillag and S. Dolfi, Semi-rational solvable groups, J. Group Theory13(4) (2010) 535–548] for semi-rational Frobenius groups.