date: Tue, 26 Mar 96 19:36:37 GMT
From: Werner Nickel 
Subject: 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 Bogley 


In 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

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