On Wed, 24 Sep 2003, Dan Hoey wrote:
I looked at your classification of groups of order 16 and verified your translation of the smaller anonymous groups I found. In hopes of identifying this group, I found its derived subgroup 2^4 .
This must be a typo for 2^2 ?
Of course G/G' is C3. But the direct product C3 x 2^4 is not G.
That seems similar to the case for A4: A4/C3 = 2^2 but 2^2x3 is not A4. Somehow I thought that was the way to decompose and reconstitute groups, but I guess I'm mistaken.
Well "this" IS the way, but of course you must get "this" right! What happens is that A4 does have a normal subgroup 2^2, but that this subgroup is not a direct factor, because the "complementary" subgroup 3 doesn't fix the 2^2 pointwise, but rather permutes its three non-identity elements cyclically. In the "ATLAS" slang it has shape 2^2:3 rather than 2^2 x 3. You might be interested to see what I regard as the best ways of thinking about all the other small groups - I went through all orders less than 32 yesterday. The hardest case, after 16, is of course 24, for which the non-abelian groups are six direct products: 2xQ12, 2xT12, 3xD8, 3xQ8, 4xD6, 2^2xD6 and six indecomposable ones D24, Q24, O24, 2T12, (12 :^5 4)/2, (4,6|2,2). Here the "tetrahedral group" T12 is another name for A(4) = 2^2:3 and the "octahedral group" O24 another name for S(4) = 2^2:D6. The group 2T12 could alternatively be described as Q8:3. The next-to-last group here is obtained from 12 :^5 4 = < a,b | 1 = a^12 = b^4, b^-1.a.b = a^5 > by identifying the central subgroups 2 of its "12" and "4", so that it has the poresentation (12 :^5 4)/2 = < a,b | above with a^6 = b^2 >. In general the group (m,n|2,2) is defined by the presentation < a,b | 1 = a^m = b^n = (ab)^2 = (a.b^-1)^2 > and this is only a sensible name when m and n are both even. Structurally, it has a normal subgroup generated by a^2 and b^2 that is the direct product of cyclic groups of orders m/2 and n/2, but I can't really find a good "structural notation" that conveys how this completes to the entire group. JHC