[GAP Forum] "(C4 x C2) : C2"

[Richard Barraclough]

Hi Nilo,

As you have noticed, the 'shape' of a group, i.e., what you get from
StructureDescription(), does not determine the isomorphism type of the
group.

There are two split extensions of (4 x 2) by 2, the action of the outer 2 on
the normal 4x2 is different.

With Ken's notation we have ( <x> x <y> ) : <z>.

Now, 4x2 has two cyclic subgroups of order 4, one generated by x that we can
see clearly, the other generated by xy. In case [16,3] z acts to swap these.

4x2 has three cyclic subgroups of order 2, generators are x^2, y and x^2y.
In case [16,3] z acts to swap <y> with <x^2 y>. Notice that this fixes the
two subgroups of order 4

You can't swap any other pair of order 2 subgroups. For example,
y -> x^2 -> xyxy = x^2 -> x^2 -> ...
which is nonsense.

It is also impossible to do both of these swaps at once: The first requires
you to swap x with xy, the second forces you to swap x with x^3y.

Therefore these are the only two groups of shape (4x2):2.


I seem to remember that "Groups for Undergraduates" by J. Moody determines
all of the groups of order up <something>. I expect there are many other
references.

Richard.


> Hi Nilo,
> Both groups are semidirect products of a normal subgroup isomorphic to
> C4 x C2 with a subgroup of order 2.
> 
> (More explicitly, according to some notes of mine, group [16,3] is
> generated by elements x, y, z where x has order 4, y and z have order
> 2, x and y commute (thus <x,y> = C4 x C2), y and z commute and zxz=xy.
> Group [16,13] is generated by x, y, z with orders 4, 2, 2, respectively
> where xy=yx, xz=zx, zyz=x^2y.)
> 
> ken
> ---
> On Feb 10, 2006, at 4:11 AM, Nilo de Roock wrote:
> 
>> Hello GAP Forum,
>> 
>> Could someone please explain why AllGroups(16)[3] and
>> AllGroups(16)[13] both return "(C4 x C2) : C2" on the function
>> StructureDescription?
>> 
>> Thanks in advance,
>> nilo
>> 
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>> 
> ---
> Ken W. Smith, Professor of Mathematics, Central Michigan University
> 989-854-0185 (Cell)
> http://www.cst.cmich.edu/users/smith1kw
> Address for 2005-06:
>      22 Chase Gayton Terrace, Apt 1518
>      Richmond, VA 23238-6526
> 
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