Isomorphism of relative holomorphs and matrix similarity

Let V be a finite dimensional vector space over the field with p elements, where p is a prime number. Given arbitrary α, β ∈ GL(V), we consider the semidirect products V ⋊ (α) and V ⋊ (β), and show that if V ⋊ (α) and V ⋊ (β) are isomorphic, then α must be similar to a power of β that generates the same subgroup as β; that is, if H and K are cyclic subgroups of GL(V) such that V ⋊ H ≈ V ⋊ K, then H and K must be conjugate subgroups of GL(V). If we remove the cyclic condition, there exist examples of nonisomorphic, let alone nonconjugate, subgroups H and K of GL(V) such that V ⋊ H ≈ V ⋊ K. Even if we require that noncyclic subgroups H and K of GL(V) be abelian, we may still have V ⋊ H ≈ V ⋊ K with H and K nonconjugate in GL(V), but in this case, H and K must at least be isomorphic. If we replace V by a free module U over Z/pmZ of finite rank, with m > 1, it may happen that U ⋊ H ≈ U ⋊ K for nonconjugate cyclic subgroups of GL(U). If we completely abandon our requirements on V, a sufficient criterion is given for a finite group G to admit nonconjugate cyclic subgroups H and K of Aut(G) such that G ⋊ H ≈ G ⋊ K. This criterion is satisfied by many groups.