On varieties of groups generated by wreath products of abelian groups

Original · EN

Generalizing results of Higman and Houghton on varieties generated by wreath products of finite cycles, we prove that the (direct or cartesian) wreath product of arbitrary abelian groups A and B generates the product variety var (A) · var (B) if and only if one of the groups A and B is not of finite exponent, or if A and B are of finite exponents m and n respectively and for all primes p dividing both m and n, the factors B[pᵏ]/B[pᵏ⁻¹] are infinite, where B[s]= b∈ B|bˢ=1 and where pᵏ is the highest power of p dividing n.