On the Exterior Degree of Finite Groups

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.

On the multiple exterior degree of finite groups

“…The importance of the condition d(G) = d(G) = d ∧ (G) is illustrated in [13] in finite case. The following result is an application of Theorem 3.7 and, at the same time, is a generalization of the corresponding results of [13] to the infinite case. For all x ∈ Z(G), arguing as in Theorem 3.7, we find that |G :…”

Commuting Powers and Exterior Degree of Finite Groups

“…In particular, [17, Lemma 2.10] can be found as a special case of the previous result. Another general property is encountered when we go to form quotients and for m = 1 it can be found in [17,Proposition 2.6]. Before to describe it, we introduce the set Z ∧ (H, K) = {h ∈ H | h ∧ k = 1 H∧K ∀k ∈ K}, where H and K are normal subgroups of G, acting upon each other by conjugation.…”

“…In [8,16,17] it was noted that a group G such that Z ∧ (G) = Z(G) has strong structural restrictions; among these it was noted in [17] that d ∧ 1 (G) = d ∧ (G) = d(G). We find something of similar in the next result.…”

“…is the m-th exterior degree of G and, of course, d ∧ 1 (G, G) = d ∧ (G) so that it is meaningful to generalize the bounds in [17]. We also note that for H = G and m = 1 there are results on d ∧ (G, K) in [18].…”

“…in [6], where H is a normal subgroup of G, K is an arbitrary subgroup of G and k K (H) is the number of the K-conjugacy classes [h] K = {h k | k ∈ K} that constitute H. We will focuse on a recent contribution in [17], where it is introduced the exterior degree of G…”