date: Tue, 26 Mar 96 19:36:37 GMT From: Werner NickelSubject: Re: non-solvable groups of order 120 Ken Smith wants to know: > How many non-solvable groups of order 120 are there? There are three non-soluble groups of order 120. There are the two groups mentioned (S_5 and C_2 x A_5) and there is SL(2,5), the group of all 2x2 matrices over GF(5) with determinant 1. It has a central involution: the negative -I of the identity matrix. SL(2,5) is perfect and therefore not isomorphic to one of the other groups. Another way to see that SL(2,5) is not isomorphic to the direct product C_2 x A_5 is to observe that the square of the matrix [ 0 1 ] [ -1 0 ] is equal to -I. > In particular, if I have a commuting > involution s and G/ is isomorphic to A_5, then is G = Z_2 x A_5? Why? It could be SL(2,5). Another way of describing SL(2,5) is as the Schur cover of A_5. All the best, Werner Nickel. --- Subject: Nonsoluble groups of order 120 Date: Tue, 26 Mar 1996 12:11:47 PST From: Bill BogleyIn addition to S_5 and A_5 x Z_2, there is the "binary icosahedral group" BI = (x,y: x^5 = y^3 = (xy)^2) which admits a permutation representation in A_5 given by x = (12345) y = (153) The element z = x^5 is a central involution in BI. The presentation makes is clear that BI is perfect (has trivial commutator quotient) and so does not exhibit Z_2 as a quotient group. Historical Note: This group was encountered (discovered?) by Poincare in the 1880/90s; BI is the fundamental group of a certain closed orientable three-manifold with the same singular homology as the three-sphere...a homology three-sphere. This led Poincare to reformulate his initial guess/question as to whether a homology three-sphere had to be homeomorphic to the three-sphere. The question now remains as to whether a homotopy three-sphere (a simply connected homology three-sphere) must be homeomorphic to the three-sphere (aka Poincare Conjecture). I now see that someone else has beaten me to it...alas and alack, but here you go anyway. Bill Bogley ============================================================================ Department of Mathematics bogley@math.orst.edu Kidder 368 tel: (503)737-5158 Oregon State University fax: (503)737-0517 Corvallis, OR 97331-4705 ftp.math.orst.edu/publications/bogley USA http://www.orst.edu/~bogleyw ============================================================================