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).