One more response to the group theory question......DMD ___________________________________________________ Subject: Group theory question: matched pairs of groups From: "Ronald Brown" Date: Thu, 31 Mar 2005 11:11:53 +0100 reply only to r.brown@bangor.ac.uk So far it has not been mentioned in correspondence on this that if G=HK, a direct product of subgroups with intersection consisting of the identity, then H,K form what is known as a `matched pair of groups'. An account of this is in S. Majid `Foundations of quantum group theory', CUP 1995, or earlier (e.g. Takeuchi Comm. Alg 9(1981) 841-882), and more recently N. Andruskiewitsch and S. Natale, JPAA, 182 (2003) 119-149. Note that for g in G we can (using g inverse) write uniquely g=hk=k'h' with h,h' in H, k,k' in K. If we write k' = {^h}k, h'= h^k then the rules for these operations are nicely pictured (see references, different notation) from a double groupoid approach where the squares are quadruples [ {^h}k ] | h h^k | [ k ] with easily deduced horizontal (to the right) and vertical (downwards) compositions. This also shows how to reconstruct G from H,K and these operations with these laws (as these references show). These double groupoids are special (also known as vacant) in that the squares are determined by two of the edges. Such double groupoids are surely too special for a homotopy theorist, who would expect double groupoids to model some 2-dimensional properties, e.g. homotopy 2-types, which they can usefully, even calculably, do. Ronnie Brown www.bangor.ac.uk/~mas010 >> Subject: group theory question >> From: "Michael Cole" >> Date: Mon, 28 Mar 2005 23:44:41 -0500 >> >> 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.