Remarks on the Relative Tensor Degree of Finite Groups

The present paper is a note on the relative tensor degree of finite groups. This notion generalizes the tensor degree, introduced recently in literature, and allows us to adapt the concept of relative commutativity degree through the notion of nonabelian tensor square. We show two inequalities, which correlate the relative tensor degree with the relative commutativity degree of finite groups. 1. The Relative Tensor Degree All the groups of the present paper are supposed to be finite. Having in mind the exponential notation for the conjugation of two elements x and y in a group G, that is, the notation xy = y−1xy, we may follow [3, 4, 17] in saying that two normal subgroups H and K of G act compatibly upon each other, if ( h1 2 )h1 = (( h2 −1 1 )k1)h1 and ( k1 2 )k1 = (( k2 −1 1 )h1)k1 for all h1, h2 ∈ H and k1, k2 ∈ K, and if H and K act upon themselves by conjugation. Given h ∈ H and k ∈ K, the nonabelian tensor product H⊗K is the group generated by the symbols h⊗ k satisfying the relations h1h2 ⊗ k1 = (h1 2 ⊗ k1 h1 ) (h1 ⊗ k1) and h1 ⊗ k1k2 = (h1 ⊗ k1) (h1 1 ⊗ k k1 2 ) for all h1, h2 ∈ H and k1, k2 ∈ K. The map κH,K : h ⊗ k ∈ H ⊗ K 7→ [h, k] = h−1hk ∈ [H,K] = 〈[h, k] | h ∈ H, k ∈ K〉 turns out to be an epimorphism, whose kernel kerκH,K = J(G,H,K) is central in H ⊗K. The reader may find more details and a topological approach to J(G,H,K) in [4, 5, 13, 17]. The short exact sequence 1 −−−−−→ J(G,H,K) −−−−−→ H ⊗ K κH,K −−−−−→ [H,K] −−−−−→ 1 is a central extension. In the special case G = H = K, we have that J(G) = J(G,G,G) = kerκG,G = kerκ and H ⊗ K = G ⊗ G is called nonabelian tensor square of G. The fundamental properties of G ⊗ G have been described in the classical paper [3], in which it is noted that κ : x⊗ y ∈ G⊗G 7→ κ(x⊗ y) = [x, y] ∈ G′ = [G,G] is an epimorphism of groups with kerκ = J(G) and 1→ J(G)→ G ⊗ G κ →G′ → 1 is a central extension. The 2010 Mathematics Subject Classification. Primary 20J99, 20D15; Secondary 20D60, 20C25.