A correspondence induced by an involution centralizing an index-two subgroup.

Given groups (G, o) and (H, *) with disjoint support, we remark that the isomorphisms (G, o) → (H, *) are in one-to-one correspondence with certain group operations on G∪H extending the operation of G and interacting nicely with the structure on H. This correspondence follows from some general properties of groups admitting an index-two subgroup N with an involution x ∈ N which centralizes N. [ABSTRACT FROM AUTHOR]