Subject: group theory question From: "Michael Cole" Date: Mon, 28 Mar 2005 23:44:41 -0500 To: This is really a pure algebra question rather than a topology question. Recall that two subgroups H,K of G are complements if HK=G and they intersect in the trivial subgroup. In general, neither of H,K need be normal; example: the Sylow 2 and 3 subgroups of the symmetric group S4. Question: If H has two complements K1 and K2 in the finite group G must they be isomorphic? That will clearly be the case if H is normal and we have the semidirect product situation, but I don't see why that would have to be true in general. If it is not true, does anybody know the simplest counter example and/or smallest group G for which a counterexample occurs. Happy Easter to all, Mike Cole