[GAP Forum] Extension

[Alexander Hulpke]

Dear Forum, Dear Sara Radfar,

> I need know how introduce extension a group such G by Z_n .For example
> how intruduce Mathieu group M_12 by Z_2  i,e, G=M_12.2.Thanks,

In general this depends on the group in question and their representation on the computer. (There also is the question of specifying the extension, as there are often multiple ones).

Given your examples, I suspect you are particularly interested in the case that G is a simple group. In this case the extension (unless its a direct product, or has parts that act trivially) will embed into the automorphism group of $G$.

Thus, for example for M12:

g:=MathieuGroup(12);
au:=AutomorphismGroup(g);
phi:=IsomorphismPermGroup(au);
a:=Image(phi);
s:=Subgroup(a,List(InnerAutomorphismsAutomorphismGroup(au),x->Image(phi,x)));
(or simply: s:=Socle(a));

Now s is M12 embeded into the automorphism group as group of permutations.
In this case [a:s]=2, so this is the only such extension. Otherwise you'd have to look for subgroups of a/s of order n.

Best,

   Alexander Hulpke




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