Commuting Powers and Exterior Degree of Finite Groups

Abstract

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.

On the multiple exterior degree of finite groups

“…Roughly speaking, Theorem 2.2 shows a passage under projective limit for the operator ∧ and ensures a topological structure on G ∧G. This allows us to generalize the exterior centralizer C ∧ E (x) of an element x of a finite group E and the exterior center Z ∧ (E), recently studied in [12,13,14,15,16,17]. The interest for C ∧ E (x) and Z ∧ (E) is due to the fact that they provide criteria to decide if we have a capable group or not (see [1,12,13]), that is, if our group is isomorphic or not to the inner automorphism of another group.…”

Remarks on the relative tensor degree of finite groups

“…H,K is the so-called Schur multiplier of the triple (G, H, K). We inform the reader that several references on the theory of the Schur multipliers of triples can be found in [4,11]. In particular, M (G, G, G) = M (G) = H 2 (G, Z) is the Schur multiplier of G, that is, the second integral homology group over G.…”

“…. Some recent papers as [11] show that it is possible to have a combinatorial approach for measuring how far a group G is from Z ∧ (G) and this is interesting, because a result of Ellis [5] characterize a capable groups by the triviality of its exterior center (i.e. : a group G is capable if G ≃ E/Z(E) for a given group E).…”

“…: a group G is capable if G ≃ E/Z(E) for a given group E). This aspect has motivated the notion of relative exterior degree [11]. It is easy to prove that d ∧ (G) = 1 if and only if G = Z ∧ (G).…”

The Probability That and Commute in a Compact Group

“…Since ν is σ-additive, so is P and therefore P( For the history of the general combinatorial issue of commuting elements in finite and compact groups, see [3,6,10,11]; for a more extensive list of references we refer to [8,Section 6] where the history is discussed explicitly. Some recent developments on the topic can also be found in [4,12,13].…”