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^2n+1, first one is the heisenberg group which has exponent p and the second one is of exponent p^2 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^2 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^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).