Nonabelian group of order 12

[Rob Beezer]

In introductory group theory, I like to be sure to expose the students
to every group of order 15 or less. As permutation groups, most of
these are easily available in Sage via cyclic permutation groups,
perhaps along with the function that builds direct products, dihedral
groups, etc. There are two gaps to fill though. Trac #7151 adds the
"quaternion group" (nonabelian, order 8). The remaining group is the
semidirect product of Z_3 by Z_4 (one presentation is <s, t; s^6 = 1,
s^3 = t^2, sts = t>).

The nonabelian group of order 4 is known in Sage as the
"KleinFourGroup".

My question: anybody know a succinct name for the above group of order
12? I've seen it listed a few places as "T" - does that have a
history?

Rob

[John H Palmieri]

Wikipedia seems to call it a "dicyclic group": see

<http://en.wikipedia.org/wiki/List_of_small_groups>
<http://en.wikipedia.org/wiki/Dicyclic_group>

The same name is given in

<http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/group12.pdf>

and kconrad does not seem to have written the wikipedia page, so there
are two independent sources.

John

[Rob Beezer]

Thanks, John. Mathworld and a couple of other lists I like to consult
didn't have this. And Conrad says the "quaternion group" is the
dicyclic group of order 8, the smallest member of this infinite family
of nonabelian groups of order 4m. Also known as "binary dihedral."
So maybe I'll just implement the whole family, and possibly the
construction will build the KleinFourGroup for m=1.

Rob

[David Joyner]

Possibly this (incomplete) page is useful:
http://www.opensourcemath.org/gap/small_groups.html
There is a book Group tables, by Thomas+Wood, which I used to compile it,
but I ran out of gas and left several entries (eg, groups of order 14)
completely blank.

>
> Rob
> >
>

[Rob Beezer]

Thanks for the info, David. I'd been looking at

http://shell.cas.usf.edu/~eclark/algctlg/small_groups.html

which appears quite similar.

I believe I've got a permutation representation of the dicyclic group
of order 4m in the symmetric group on m+4 symbols. But the group must
have an element of order 4, so it won't build the KleinFourGroup when
m=1.

I may be tempted to do some more work on permgroup_named.py once I'm
done with this - thanks for all your work getting those constructions
together.

Rob

[David Joyner]

On Sat, Oct 17, 2009 at 12:20 AM, Rob Beezer <goo...@beezer.cotse.net> wrote:
>
> Thanks for the info, David.  I'd been looking at
>
> http://shell.cas.usf.edu/~eclark/algctlg/small_groups.html
>
> which appears quite similar.

The story is this: I had a page, like Clark's but over a smaller range
and with a bit
less info about each group, derived form the Thomas+Wood book. Then I discovered
Clark's page and emailed around trying to find out if I could combine
them. Clark has
disappeared and the webmaster maintaining the page indicated it was now the
University's property and that could combine the two. Actually, both mine and
Clark's are simply a recitation of well-known facts in an unoriginal
format, so I
don't think copyright is an issue. In fact, my plan was to write a
Sage script to
generate the data. I ran out of gas. If/when I teach a course on group
theory again,
I'll return to it.

[Jaap Spies]

David Joyner wrote:
> On Sat, Oct 17, 2009 at 12:20 AM, Rob Beezer <goo...@beezer.cotse.net> wrote:
>> Thanks for the info, David. I'd been looking at
>>
>> http://shell.cas.usf.edu/~eclark/algctlg/small_groups.html
>>
>> which appears quite similar.
>
>
> The story is this: I had a page, like Clark's but over a smaller range
> and with a bit
> less info about each group, derived form the Thomas+Wood book. Then I discovered
> Clark's page and emailed around trying to find out if I could combine
> them. Clark has
> disappeared and the webmaster maintaining the page indicated it was now the

To my knowing Edwin Clark is still active. See his homepage:
http://shell.cas.usf.edu/~eclark/
and this citation from an e-mail:

>> PS My mathematics work seems to be tapering off these days. But I have joined the Maple 13 beta testing team and am pleased to see some nice new features. But from the sidelines I am pulling for SAGE to surpass Maple and Mathematica!
>>

He is promoting Sage!

> University's property and that could combine the two.

[...]

Jaap

[David Joyner]

On Sat, Oct 17, 2009 at 11:53 AM, Jaap Spies <j.s...@hccnet.nl> wrote:
>
> David Joyner wrote:
>> On Sat, Oct 17, 2009 at 12:20 AM, Rob Beezer <goo...@beezer.cotse.net> wrote:
>>> Thanks for the info, David.  I'd been looking at
>>>
>>> http://shell.cas.usf.edu/~eclark/algctlg/small_groups.html
>>>
>>> which appears quite similar.
>>
>>
>> The story is this: I had a page, like Clark's but over a smaller range
>> and with a bit
>> less info about each group, derived form the Thomas+Wood book. Then I discovered
>> Clark's page and emailed around trying to find out if I could combine
>> them. Clark has
>> disappeared and the webmaster maintaining the page indicated it was now the
>
> To my knowing Edwin Clark is still active. See his homepage:

Yes, sorry. I meant to say Pederson (who wrote the small groups page).
Clark moved Pederson's page to his website when Pederson disappeared.

[Rob Beezer]

On Oct 17, 5:36 am, David Joyner <wdjoy...@gmail.com> wrote:
> The story is this: I had a page, like Clark's but over a smaller range
> and with a bit
> less info about each group, derived form the Thomas+Wood book.

Alright, that's much more complicated than my initial suspicion. ;-)

It'd be great to have something more comprehensive right in Sage that
either listed in the manual a table of small groups and how to get
them from Sage, or produced output sort of like you can get from GAP's
database (but without having to add an spkg for the database). Yes,
I'll add something like this to my list, too.

[Rob Beezer]

There's an implementation of the dicyclic groups up now at

http://trac.sagemath.org/sage_trac/ticket/7244

Rob

[David Joyner]

On Mon, Oct 19, 2009 at 1:18 AM, Rob Beezer <goo...@beezer.cotse.net> wrote:
>
> There's an implementation of the dicyclic groups up now at
>
> http://trac.sagemath.org/sage_trac/ticket/7244

This is the generalized quaternion group.
http://en.wikipedia.org/wiki/Quaternion_group
(The presentation is sligtly different but I think you can derive
easily your presentation from the one given there.)

If you agree, do you think it is reasonable to add an alias
or at least to add a comment in the docstring?

Sorry for not pointing this out before, it's been busy...


>
> Rob
> >
>

[Rob Beezer]

Hi David,

Thanks for the comments. The "Quaternion Group" Wikipedia page seems
to differ substantially with the "Dicyclic Group" page:

http://en.wikipedia.org/wiki/Dicyclic_group

which says:

"More generally, when n is a power of 2, the dicyclic group is
isomorphic to the generalized quaternion group."

Its not obvious to me that presentation in the Quaternion page
guarantees that the group will have order 4n, as claimed there, but
perhaps that follows. If so, then as you have noted, it is identical
to what is called the dicyclic group on the other page.

The docstring I have in the patch right now says:
'When the order of the group is a power of 2 it is known as a
"generalized quaternion group." '

If folks agree that the term "generalized quaternion group" just
applies when the order of the group is a power of 2, then I could make
a derived class for this, like I did for just the dicyclic group of
order 8, which I just call *the* quaternion group. I thought of doing
that, but it felt like overkill, and just went with comment in the
docstring.

Thanks,
Rob

[David Joyner]

On Wed, Oct 21, 2009 at 1:35 AM, Rob Beezer <goo...@beezer.cotse.net> wrote:
>
> Hi David,
>
> Thanks for the comments.  The "Quaternion Group" Wikipedia page seems
> to differ substantially with the "Dicyclic Group" page:
>
> http://en.wikipedia.org/wiki/Dicyclic_group
>
> which says:
>
> "More generally, when n is a power of 2, the dicyclic group is
> isomorphic to the generalized quaternion group."
>
> Its not obvious to me that presentation in the Quaternion page
> guarantees that the group will have order 4n, as claimed there, but
> perhaps that follows.  If so, then as you have noted, it is identical
> to what is called the dicyclic group on the other page.
>
> The docstring I have in the patch right now says:
> 'When the order of the group is a power of 2 it is known as a
> "generalized quaternion group." '

I think you are right and I am wrong. Most people in the literature say that a
generalized quaternion group is a 2-group, so what you have
doesn't need any change after all. The presentations are the same
which I guess is what made me think they are the same.

Sorry for the noise:-)

[David Joyner]

I gave it a positive review.