Orbits in Extra-special $p$-Groups for $p$ an Odd Prime

C P Anil Kumar, Soham Swadhin Pradhan

Published 2019-08-01Version 1

For an odd prime $p$ and a positive integer $n$, it is well known that there are two types of extra-special $p$-groups of order $p^{2n+1}$, first one is the Heisenberg group which has exponent $p$ and the second one is of exponent $p^2$. In this article, a new way of representing the extra-special $p$-group of exponent $p^2$ is given, which is suggested in a natural way by a familiar representation of the Heisenberg group. These representations facilitate an explicit way of finding formulae for any automorphism and any endomorphism of an extra-special $p$-group $G$ for both the types. Based on these formulae, the automorphism group $Aut(G)$ and the endomorphism semigroup $End(G)$ are described. The orbits under the action of the automorphism group $Aut(G)$ are determined. In addition, the endomorphism semigroup image of any element in $G$ is found. As a consequence it is deduced that, under the notion of degeneration of elements in $G$, the endomorphism semigroup $End(G)$ induces a partial order on the automorphism orbits when $G$ is the Heisenberg group and does not induce when $G$ is the extra-special $p$-group of exponent $p^2$.