Let $X$ and $Y$ be connected Cayley graphs on abelian groups, such that no two distinct vertices of have exactly the same neighbours, is true about $Y$. We show if number relatively prime to $Y$, then direct product $X \times Y$ has only obvious automorphisms (namely, ones come from its factors $Y$). This was not previously known even in special case where $Y = K_2$ vertices. The proof this sho...

Definition. Let G and H be groups. Their direct product is the group G×H defined as follows. As a set G×H = {(g, h) : g ∈ G, h ∈ H} is just the usual Cartesian product of G and H (the set of ordered pairs where the first component lies in G and the second component lies in H). The group operation on G×H is defined by the formula (g1, h1)(g2, h2) = (g1g2, h1h2) for all g1, g2 ∈ G and h1, h2 ∈ H....

We study properties of a group, abelian group, ring, or monoid B which (a) guarantee that every homomorphism from an infinite direct product ∏ I Ai of objects of the same sort onto B factors through the direct product of finitely many ultraproducts of the Ai (possibly after composition with the natural map B → B/Z(B) or some variant), and/or (b) guarantee that when a map does so factor (and the...

Let G be a finite group, F a free group of finite rank, R the kernel of a homomorphism tp of F onto G, and let [ i i , f] , [R,R] denote mutual commutator subgroups. Conjugation in F yields a G-module structure on R/[R,R]; let da[R/[R, R]) be the number of elements required to generate this module. Define d(R/[R, F]) similarly. By an earlier result of the first author, for a fixed G, the differ...