# On the Endomorphism Semigroups of Extra-special $p$-groups and Automorphism Orbits

## TL;DR

This paper provides explicit formulas for endomorphisms and automorphisms of certain extra-special p-groups, describes their structure, and explores the induced partial order and combinatorial properties related to these groups.

## Contribution

It introduces a new representation for the exponent p^2 case, enabling explicit formulas for endomorphisms and automorphisms, and analyzes their algebraic and combinatorial properties.

## Key findings

- Explicit formulas for endomorphisms and automorphisms of extra-special p-groups.
- Description of the endomorphism semigroup and automorphism group structures.
- Demonstration of the partial order induced by endomorphisms on automorphism orbits.

## Abstract

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. These representations facilitate an explicit way of finding formulae for any endomorphism and any automorphism of an extra-special $p$-group $G$ for both the types. Based on these formulae, the endomorphism semigroup $End(G)$ and the automorphism group $Aut(G)$ are described. The endomorphism semigroup image of any element in $G$ is found and the orbits under the action of the automorphism group $Aut(G)$ are determined. 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 $p^2$. Finally we prove that the cardinality of isotropic subspaces of any fixed dimension in a non-degenerate symplectic space is a polynomial in $p$ with non-negative integer coefficients. Using this fact we compute the cardinality of $End(G)$.