  We calculate the mod-two cohomology of all alternating groups together, with
both cup and transfer product structures, which in particular determines the
additive structure and ring structure of the cohomology of individual groups.
We show that there are no nilpotent elements in the cohomology rings of
individual alternating groups. We calculate the action of the Steenrod algebra
and discuss individual component rings. A range of techniques is needed: an
almost Hopf ring structure associated to the embeddings of products of
alternating groups, the Gysin sequence relating the cohomology of alternating
groups to that of symmetric groups, Fox-Neuwirth resolutions, and restriction
to elementary abelian subgroups.
