on co-maximal subgroup graph of z_n$

For an odd prime p and a positive integer n, it is well known that there are two types ofextra-special p-groups of order p^2n+1, first one is the heisenberg group which has exponent p and thesecond one is of exponent p^2. this article mainly describes the endomorphism semigroups of both thetypes of extra-special p-groups and computes their cardinalities as polynomials in p for each n. firstlya new way of representing the extra-special p-group of exponent p2is given. using the representations,explicit formulae for any endomorphism and any automorphism of an extra-special p-group g forboth the types are found. based on these formulae, the endomorphism semigroup end(g) and theautomorphism group aut(g) are described. the endomorphism semigroup image of any element ing is found and the orbits under the action of the automorphism group aut(g) are determined. as aconsequence it is deduced that, under the notion of degeneration of elements in g, the endomorphismsemigroup end(g) induces a partial order on the automorphism orbits when g is the heisenberg groupand does not induce when g is the extra-special p-group of exponent p^2. finally we prove that thecardinality of isotropic subspaces of any fixed dimension in a non-degenerate symplectic space is apolynomial in p with non-negative integer coefficients. using this fact we compute the cardinality ofend(g).