We consider the problem of testing isomorphism of groups of order n given by Cayley tables. The trivial nlogn bound on the time complexity for the general case has not been improved over the past four decades. Recently, Babai et al. (following Babai et al. in SODA 2011) presented a polynomial-time algorithm for groups without abelian normal subgroups, which suggests solvable groups as the hard ...

Let p be a prime,G a finite group which has a normal p-subgroup containing its own centralizer inG, and R a commutative local ring with residue class field of characteristic p. In this paper, it is shown that if α is an augmented automorphism of RG which fixes a Sylow p-subgroup P of G, there is ρ ∈ Aut(G) such that xαρ = x for all x ∈ P , and αρ is an inner automorphism of RG.

A complete Sylow sequence, P = P1, . . . , Pm , of a finite group G is a sequence of m Sylow pi -subgroups of G, one for each pi , where p1, . . . , pm are all of the distinct prime divisors of |G|. A product of the form P1 · · · Pm is called a complete Sylow product of G. We prove that the solvable radical of G equals the intersection of all complete Sylow products of G if, for every compositi...