Groups isomorphic via decomposition

Abstract

into two parts: G = G, U G2 and H = HI U H, with G, - HI and G2 - H,. The notation A - B, where A c G and B c H, means there is a one to one map f of A onto B such that if x, y, and xy are all in A, then f(xy) = f(x)f(y). The purpose of this note is to give an elementary set of examples for groups G and H. Let G be any group with a normal subgroup N of index 2. Now G = N U gN for any g $Z N, and N = N U N. Clearly N - N via the identity map. We also have gN - N via the map f: gN + N defined by f (gn) = n. f is obviously one to one and onto. Now G/N is a cyclic group of order two, and thus (gN )(gN) = N, and so if x and y are elements of gN then xy E N. Thus the condition f (xy) = f(x) f ( y) if x, y, and xy are all in gN, is vacuously satisfied. The answer to the question is affirmative and just as easy if we further require both decompositions to be partitions. To construct examples pick any group N such that you can find two non-isomorphic extensions of N by a cyclic group of order two. Again these two extensions will work as examples of G and H. The partitions into the cosets of N will be the required decompositions. A specific example of this is Q, and D,, the quatemions and the dihedral group of order 8. Of course, in both cases by changing the index 1 G : Nl, we can find examples of groups “isomorphic via decomposition” into 1 G : N 1 parts.