Re: Groups of order 36, 40, 56

From: Jim Heckman (wnzrfeurpxzna_at_lnubb.pbz.invalid)
Date: 02/02/05 

Date: Wed, 2 Feb 2005 07:36:59 GMT

On 1-Feb-2005, "Van" <calccurve-spam123@yahoo.com>
wrote in message <1107258465.335757.226320@z14g2000cwz.googlegroups.com>:

[...]

> As for groups of order 36, the 2 coming from C_2 x C_2 acting on C_3 x
> C_3 being normal,

Um, there are *4* such groups, as per my previous post.

> must be S_3 x S_3 = (C_3 x| C_2) x (C_3 x| C_2)
> with 4 generators a,b,x,y with a^2 = b^2 = 1 = x^3 = y^3 and ab = ba,
> xy = yx,

Don't forget the relations axax, [a,y], [b,x] and byby, if
you're making everything explicit. (I usually leave trivial
commutators out of my own presentations, so I would typically
list only axax and byby for this group.)

> and Z_6 X S_3.

You left out the abelian group C_3 x C_3 x C_2 x C_2, and the
group ((C_3 x C_3) x| C_2) x C_2 = <x,y,a,b | a^2, b^2, x^3,
y^3, axax, ayay>. Now *why* are these the only 4 groups with
C_2 x C_2 acting on C_3 x C_3?

> There is also the other abelian C_2 x C_2 x C_9, etc.,
> actual there must be 4 abelian groups, 2 with C_9 and 2 with C_3 x C_3,
> right?

Yes. So why did you leave out one of the ones with C_3 x C_3
just above? :-/ 

-- 
Jim Heckman
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