We study the number of elements x and y of a finite group G such that x⊗ y = 1G⊗G in the nonabelian tensor square G⊗G of G. This number, divided by |G|2, is called the tensor degree of G and has connection with the exterior degree, introduced few years ago in [P. Niroomand and R. Rezaei, On the exterior degree of finite groups, Comm. Algebra 39 (2011), 335-343]. The analysis of upper and lower bounds of the tensor degree allows us to find interesting structural restrictions for the whole group.
AUTHOR KEYWORDS: Commutativity degree; Dihedral groups; Exterior degree; P-groups; Schur multiplier; Tensor degree PUBLISHER: Charles Babbage Research Centre
The concept of exterior degree of a finite group C is introduced by the author in a joint paper  which is the probability of randomly two elements g and h in G such that gδ = 1. In the present paper, a necessary and sufficient condition for a non cyclic group is given when its exterior degree achieves the upper bound (p2 + p - 1)/p3 in which p is the smallest prime number dividing the order of C. We also compute the exterior degree of all extra-special p-groups. Finally, for an extra-special p-group H and a group C when G/Zδ (G) is p-group, we will show that dδ (C) = dδ (H) if and only if G/ZA(G)∼H/Zδ (H) provided that dA(C) 11/32.
AUTHOR KEYWORDS: Capable group; Schur multiplier; extra-special group.; Commutativity degree; Exterior centre; Exterior degree; Phrases PUBLISHER: Charles Babbage Research Centre
Let G be a finite p-group of order pn. It is known that |M(G)| = p1/2n(n-1)-t(G) and t(G) ≥ 0. The structure of G for t(G) ≥ 4 was determined by several authors. In this paper we will describe all the possible structures of G for t(G) = 5.
AUTHOR KEYWORDS: P-groups; Schur multiplier PUBLISHER: Editura Academiei Romane
An improvement of a bound of Yankosky (2003) is presented in this paper, thanks to a restriction which has been recently obtained by the authors on the Schur multiplier M(L) of a finite dimensional nilpotent Lie algebra L. It is also described the structure of all nilpotent Lie algebras such that the bound is attained. An important role is played by the presence of a derived subalgebra of maximal dimension. This allows precision on the size of M(L). Among other results, applications to the non-abelian tensor square L ⊗ L are illustrated.
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