Let G(H) be the monoid of H
equivariant self maps of S(nV ), the unit sphere of n copies of a finite dimension
orthogonal representation V of a finite group H, stabilized over n in an
appropriate way. Let SG(H) be the submonid of G(H) consisting of all
degree 1 maps. If H1 is a subgroup of H there is a natural forgetful map
SG(H) → SG(H1) and if Z is the center of H there is a natural action map
BZ × SG(H) → SG(H) induced by the natural action of Z on H. The
main results of this paper are the calculations of the Hopf algebra structures
of H∗(SG(Z∕pn),Z∕p) and H∗(BSG(Z∕pn),Z∕p) for all n and all primes
p, the calculations in homology of forgetful maps induced by the natural
inclusions Z∕pn−1→ Z∕pn and, for H = Z∕2, the calculation of the action map
H∗(RP∞,Z∕2)⊗H∗(BSG(Z∕2),Z∕2) → H∗(BSG(Z∕2),Z∕2).
