On the endomorphism semigroups of extra-special p-groups and automorphism orbits

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^{۲n+۱}, first one is the Heisenberg group which has exponent p and the second one is of exponent p^۲. This article mainly describes the endomorphism semigroups of both the types of extra-special p-groups and computes their cardinalities as polynomials in p for each n. Firstly a new way of representing the extra-special p-group of exponent p^۲ is given. Using the representations, explicit formulae for any endomorphism and any automorphism of an extra-special p-group G for both the types are found. 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^۲. 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).

Keywords:

Extra-special p-Groups , Heisenberg Groups , Automorphism Groups , Endomorphism Semigroups , symplectic groups