No abstract

Let M be a normal subgroup of a finite group G such that |G/M| = pm and |M/Z(M,G)| = pn. In this paper, we characterize all the pairs of groups (M,G) which satisfy the conditions [Formula: see text] and |[M/Z(M,G),G/Z(M,G)]|≤ p. If M has a complement in G and |M|=pn and |G/M|=pm, then there always exists a non-negative integer t(M,G) such that [Formula: see text]. Under some conditions, we determine all the pairs (M,G) with t(M,G)=0, 1 or 2.

This paper deals with the Choi's inequality for measurable operators affiliated with a given von Neumann algebra. Some Young and Cauchy-Schwarz type inequalities for-measurable operators are also given.