No abstract

We introduce the exterior degree of a finite group G to be the probability for two elements g and g in G such that g ∧ g = 1, and we shall state some results concerning this concept. We show that if G is a non-abelian capable group, then its exterior degree is less than 1/p, where p is the smallest prime number dividing the order of G. Finally, we give some relations between the new concept and commutativity degree, capability, and the Schur multiplier.

Abstract. Recently, we have introduced a group invariant, which is related to the number of elements x and y of a finite group G such that x ∧ y = 1 G∧G in the exterior square G ∧ G of G. This number gives restrictions on the Schur multiplier of G and, consequently, large classes of groups can be described. In the present paper we generalize the previous investigations on the topic, focusing on the number of elements of the form h m ∧ k of H ∧ K such that h m ∧ k = 1 H∧K , where m ≥ 1 and H and K are arbitrary subgroups of G.

The exterior degree of a pair of finite groups (G, N ), which is a generalization of the exterior degree of finite groups, is the probability for two elements (g, n) in (G, N ) such that g ∧ n = 1. In the present paper, we state some relations between this concept and the relative commutatively degree, capability and the Schur multiplier of a pair of groups.

P. Hall introduced the notion of isoclinism between two groups more than 60 years ago. Successively, many authors have extended such a notion in different contexts. The present paper deals with the notion of relative n-isoclinism, given by N. S. Hekster in 1986, and with the notion of n-th relative nilpotency degree, recently introduced in literature.