Commuting powers and exterior degree of finite groups. (Q2898905)

Let \(G\) be a finite group and \(H\) and \(K\) be normal subgroups. Write \(^gx:=gxg^{-1}\) for \(g\in G\) and \(x\) in \(H\) or \(K\). The exterior product \(H\wedge K\) is defined to be the group generated by the symbols \(h\otimes k\) (\(h\in H\), \(k\in K\)) subject to the relations NEWLINE\[NEWLINEhh'\otimes k=(^hh'\otimes{^kk})(h\otimes k),\quad h\otimes kk'=(h\otimes k)(^kh\otimes{^kk'})\quad\text{and}\quad y\otimes y=1,NEWLINE\]NEWLINE where \(h,h'\in H\), \(k,k'\in K\) and \(y\in H\cap K\). The image of \(h\otimes k\) in \(H\wedge K\) is denoted by \(h\wedge k\). For each integer \(m\geq 1\) the authors define NEWLINE\[NEWLINEd_m^{\wedge}(H,K)=\frac{|\{(h,k)\in H\times K\mid h^m\wedge k=1\}|}{|H\times K|}NEWLINE\]NEWLINE (the \(m\)-th relative exterior degree). In a previous paper [see \textit{P. Niroomand} and \textit{F. Russo}, Arch. Math. 93, No. 6, 505-512 (2009; Zbl 1205.20039)] two of the authors studied properties and bounds on \(d^{\wedge}(G):=d_1^{\wedge}(G,G)\). The present paper shows that similar results are true for \(d_m^{\wedge}(H,K)\) more generally.