Reformulation: non split extensions of finite groups

[Marc Bogaerts]

I have reformulated a request I made earlier a few days ago:

Could anyone provide me with an example of an exact sequence of finite
groups:
1 -> N -> G -> Q -> 1
that is
1) non split
2) N is not abelian.
3) and N is a minimal normal subgroup of G.

I would like to remark that solvable groups are excluded because a
minimal normal subgroup of a solvable group is always abelian.
Obviously simple groups G are excluded too.

[Jim Heckman]

On 14-May-2011, Marc Bogaerts <mbg-do...@gmail.com>
wrote in message <AItzp.510$Wy2...@newsfe23.ams2>:

1 -> A_6 ~= PSL(2,9) -> M_{10} -> C_2 -> 1

where M_{10} < Aut(A_6) ~= PGammaL(2,9) is the so-called Mathieu
group of degree 10, the point stabilizer of the sporadic simple
group M_{11} in its faithful action of minimal degree. Out(A_6) =
C_2 x C_2, so Aut(A_6) has two other subgroups of index 2, namely
S_6 and PGL(2,9), but these both split over A_6.)

--
Jim Heckman

[Marc Bogaerts]

Nice example indeed. The question arises whether this is the only
example, or are there more of them.

[Jim Heckman]

On 19-May-2011, Marc Bogaerts <mbg-do...@gmail.com>
wrote in message <zbnBp.4389$ue1....@newsfe28.ams2>:

According to the _ATLAS of Finite Groups_, there is what appears to
be a very similar case with N = PSL(2,25), G = N.{2_3} and Q = C_2.
Every element in G\N has order 4, 8 or 12, just as every element in
M_{10}\A_6 has order 4 or 8. Here again of course Out(N) = C_2 x
C_2 (generated by the diagonal and field automorphisms), with the
other two subgroups of Aut(N) of index 2, both of which split over
N, being N.{2_1} = PGL(2,25) and N.{2_2} = PSigmaL(2,25). (Recall
that S_6 ~= PSigmaL(2,9).)

I don't know this area at all, but it wouldn't surprise me if this
happens for all PSL(2,p^2) with odd p, maybe even for all even
powers of odd p. For such prime powers there is a family of groups
M(q) (note that M(9) = M_{10}), which are known to be the only
faithful sharply 3-transitive (of degree q) finite groups other than
PGL(2,q) (for all prime powers q). Maybe all of the M(q)'s are
non-split extensions of PSL(2,q)'s?

For an introduction to the M(q)'s, see the section "Sharply
3-Transitive Groups" of Chapter 9 /Permutations and the Mathieu
Groups/ of Joseph J. Rotman's _An Introduction to the Theory of
Groups_ (4/e 1995 Springer-Verlag).

--
Jim Heckman

[Marc Bogaerts]

Sorry for the delay, but I have been digging somewhat deeper on the
subject and found the following conclusion.

In order for a finite group N to define an extension N->G->C_2 it is
necessary and sufficient that:
1) there is an automorphism t on N.
2) there is an element g in N such that g^t=g and n^t²=n^g for all n in N.
The latter is equivalent to saying that t^2 is an inner automorphism
corresponding to the element g of N, so of the form g*e where e is an
element of the centralizer of N.

If these conditions are fulfilled then the following operation on the
cartesian product [-1,1] x N defined by:
(-1,n) ° (-1,m) = (1, g*n^t*m),
(-1,n) ° (1,m) = (-1, n*m),
(1,n) ° (-1,m) = (-1, n^t*m),
(1,n) ° (1,m) = (1, n*m)
is an associative law and defines a group structure. Inversely, if such
a product is associative then the two conditions mentioned above are
satisfied.

Now, for an element of the form (-1,n) to have order 2 we must have
g*n^t*n = 1, or equivalently, (n*t^-1)^2=e (1) for some n in N. Now
the existence of a solution to (1) only depends on the coset of t.
My findings up to now are that for the first powers of an odd prime
there are g and t such that the constructed extension by SL(2,p) is
non-split. These are the things I found:

g t
SL(2,5) [[3,0],[0,2]] [[3,0],[0,1]]
SL(2,7) [[5,0],[0,3]] [[5,0],[0,1]]
SL(2,9) [[z^5,0],[0,z^3]] [[z^5,0],[0,1]]
SL(2,11) [[7,0],[8,0]] [[7,0],[0,1]]

Why there are non-split extensions by PSL(2,p^2) is still a mystery to me.