Some Results on the Exterior Degree of Extra-Special Groups

Abstract

The concept of exterior degree of a finite group \(G\) is introduced by the author in a joint paper [13], which is the probability of randomly selecting two elements \(g\) and \(h\) in \(G\) such that \(g\wedge h = 1\). In the present paper, a necessary and sufficient condition is given for a non-cyclic group when its exterior degree achieves the upper bound \((p^2 + p – 1)/p^3\), where \(p\) is the smallest prime number dividing the order of \(G\). We also compute the exterior degree of all extra-special \(p\)-groups. Finally, for an extra-special \(p\)-group \(H\) and a group \(G\) where \(G/Z^\wedge(G)\) is a \(p\)-group, we will show that \(d^\wedge(G) = d^\wedge(H)\) if and only if \(G/Z^\wedge(G) \cong H/Z^\wedge(H)\), provided that \(d^\wedge(G) \neq 11/32\).