Re: Group of order 192

On Jan 21, 4:46 am, Derek Holt <ma...@xxxxxxxxxxxxx>
wrote:
On 21 Jan, 06:37, Jack Schmidt
<Jack.Schmidt.SciM...@xxxxxxxxx> wrote:



Conceivably the group of order 192 that I
thought
of is a realization of this, but I'm not sure.

The group of order 192 I came up with is a
subgroup
of the special linear group of 2x2 matrices
over Z24,
namely the upper triangular matrices:

[ a b ]
[ 0 c ]

having determinant ac = 1. Clearly a,c must
belong
to the residues {1,5,7,11,13,17,19,23} coprime
to 24,
i.e. the multiplicative group Z24*, and the
value of
c is determined by the choice of a (or
conversely).

So the order of this group is 8 times 24
(number of
possible b entries), or 192.

They are not quite the same. For a ring R, let
R^* be
its group of units, and let R=Z/24Z. Your group
is the
semi-direct product R^* |x R, with R^* acting on
the
normal subgroup R. Similarly AGL(1,R)=Aff(R) is
the
semi-direct product of R^* |x R, with R^* acting
on R,
but the two actions are different.

In your group R^* is embedded with a -> [a,0;0,c]
where
ac=1, and R is embedded as b->[1,b;0,1]. Then
for a in
R^* and b in R, the image of a takes the image of
b to
the image of a^2*b under conjugation in your
group.

And since a^2 = 1 for all a, this group is abelian
and isomorphic to
C2xC2xC2xC24.

However, in AGL(1,R), the image of a takes the
image
of b to the image of a*b.

So this group is not abelian.

Derek Holt.



A matrix version of AGL(1,R) is:
{ [a,b;0,1] : a in R^*, b in R }.

Very illuminating comments, Jack and Derek. Thanks!

--c

Yes, thanks everyone. This is really interesting. Just a question, does the work I did in GAP seem correct? (A few comments back).

PGH
.

Relevant Pages