The Schur–zassenhaus Theorem

When N is a normal subgroup of G, can we reconstruct G from N and G/N? In general, no. For instance, the groups Z/(p2) and Z/(p) × Z/(p) (for prime p) are nonisomorphic, but each has a cyclic subgroup of order p and the quotient by it also has order p. As another example, the nonisomorphic groups Z/(2p) and Dp (for odd prime p) have a normal subgroup that is cyclic of order p, whose quotient is cyclic of order 2. If we impose the condition that N and G/N have relatively prime order, then something nice can be said: G is a semidirect product of N and G/N . This is the Schur-Zassenhaus theorem, which we will discuss below. It doesn’t uniquely determine G, as there could be several non-isomorphic semi-direct products of the abstract groups N and G/N , but each one is a group with normal subgroup N and quotient by it isomorphic to G/N . For instance, if N ∼= Z/(p) for odd prime p and G/N ∼= Z/(2) then G must be a semi-direct product Z/(p) o Z/(2). The only two semidirect products are the direct product (which is isomorphic to Z/(2p)) and the nontrivial semidirect product (which is isomorphic to Dp).