Talk:List of small groups

Dead link
The reference at the end of the page is a dead link. Would anybody fix it?--131.215.134.104 21:09, 2 August 2007 (UTC)

The dead link to Small Groups library is now fixed. TristramBrelstaff (talk) 10:27, 21 October 2008 (UTC)

Dn
What is Dn when n is odd?? I mean, given you're using the 2n-convention. What is D3, for instance? Is it just the cyclic group? What else could it be? It must have exactly 3 elements, if D3 &times; C2 is really isomorphic to D6. But then it must be C3. The ordinary definition doesn't make sense. We can't let one element have order "1.5" and the other order 2. Revolver 02:25, 10 Sep 2004 (UTC)


 * It looks as if these were added by someone using the other convention, where Dn is the dihedral group of order 2n. The isomorphisms given (e.g., D6 = D3 &times; C2) then make sense, though they are for groups of twice the stated size. But as it stands, they make no sense at all, so I'm removing them. --Zundark 10:32, 31 Oct 2004 (UTC)

Q8 &times; Z2
The cycle graph of Q8 &times; Z2 is wrong. The number of circles is 20 and the unit isn't marked. (I don't know how to correct it.)

The correct cycle graph looks like the one for the Pauli matrices, but with six elements on each size above and only two tails below (there are 12 elements of order 4, all of which have the same square, and there are two additional elements of order 2). But I also do not know how to correct the drawing.

Notation
Considering the confusion it would be better to use, at least in Wikipedia, a uniform notation. Is Dn for order 2n more common?--Patrick 12:27, 5 August 2005 (UTC)

Cycle graphs for order 16
If anyone can supply product tables for those three missing groups of order 16, I will draw up cycle graphs for them. PAR 03:36, 2 Apr 2005 (UTC)

Product tables are cumbersome, but I can tell you what the operations are. G(4,4) is the group of pairs of integers modulo 4 with the operation
 * $$(a,b)*(c,d) = (a + (-1)^{bc}c, d + (-1)^{bc}b)$$

The generalized quaternion group is generated by the matrices
 * $$\begin{bmatrix}e^{i\pi/4} & 0 \\ 0 & e^{-i\pi/4}\end{bmatrix},\qquad

\begin{bmatrix}0 & 1 \\ -1 & 0 \end{bmatrix}$$ I don't know which groups you mean by x3 and x4. Judging from the cycle graph you given I'm guessing x3 is the semidirect product of C4 with C4, which means x4 must be the group generated by the Pauli matrices. -- Fropuff 07:03, 2005 Apr 2 (UTC)

Error in cycle graph of Dih_4xZ_2?
Isn't there an edge missing from the "topmost" element to the neutral element? (June 19, 2006)


 * There would be an edge there if we were drawing all the cycles, but we are not. From Cycle graph (algebra): Cycles which contain a non-prime number of elements will implicitly have cycles which are not connected in the graph. For the group Dih4 above, we might want to draw a line between a2 and e since (a2)2=e  but since a2 is part of a larger cycle, this is not done. —Keenan Pepper 04:27, 20 June 2006 (UTC)

About the same group, I think there should be 11 copies of Z_2^2, not 7. Can someone confirm this? Thehotelambush (talk) 00:07, 8 April 2008 (UTC)

Format error in the table for order 16
The table for groups of order 16 has a format error. The last line is misaligned. Albmont 11:43, 3 May 2007 (UTC)


 * Fixed. Someone had messed up the template that this page uses. --Zundark 11:55, 3 May 2007 (UTC)

Cycle diagram for $Z 4 2$
Shouldn't there be 3 pairs of 4-cycles which share the square element? i.e. take the diagram for Z4 &times; Z2, cut off its two "legs", make three copies, and glue them together at the identity. --192.75.48.150 21:00, 14 May 2007 (UTC)


 * That's better, thanks. --192.75.48.150 20:44, 12 June 2007 (UTC)

Small dimensional groups
How about a contribution on "Small dimensonal groups" being on those with small dimensional faithful representations in standard characteristics? This would be useful to a large community of users. John McKay24.200.155.110 (talk) 08:12, 14 August 2008 (UTC)


 * For wikipedia: Do you know of such a published list? There are lists of maximal irreducible soluble subgroups over finite fields (in order to get the soluble primitive permutation groups), and there are lists of maximal subgroups of classical groups for small dimension (maybe into the two digits now, but perhaps only up to 8 or so).
 * For curiosity: How would one decide what groups to list for a given dimension d and characteristic p? Even for d=1, one has infinitely many finite cyclic groups for each p, and higher d are harder to describe without resorting to "and its subgroups". JackSchmidt (talk) 12:40, 15 August 2008 (UTC)

There's not too much reason to list cyclic, or even abelian groups. A group theorist could write any needed representation on demand. Anyone else ... is unlikely to care much. But it would be interesting to list representations of non-abelian groups of common types (the linear groups, for example, and perhaps p-group representations for smallish orders and common families. In particular, if the characteristic of the representing field doesn't divide the group order, the representation is likely to be fairly tractable and sometimes interesting. Erbach (talk) 18:54, 4 May 2014 (UTC)

Missing cycle graphs
Why no cycle graphs for Z3 = A3 and S3 = Dih3? I'd add them if I knew how... LaQuilla (talk) 13:10, 8 November 2008 (UTC)

Modular group
What is meant by "The order 16 modular group"? The page linked doesn't describe a finite group at all. --Octavo (talk) 12:15, 11 January 2009 (UTC)


 * I think it's meant in the sense of M-group, in which case the link should be changed. The group in question seems to be the one with exactly two non-normal subgroups, and in this group all subgroups are permutable, so the subgroup lattice is modular. --Zundark (talk) 16:13, 11 January 2009 (UTC)

How many groups of order 1024?
This article says, under "Small group library": "except for order 1024 (423164062 groups; the ones of order 1024 had to be skipped, there are alone 49487365422 nonisomorphic 2-groups of order 1024.)," which seems inconsistent on its face as the first number is smaller than the second. Our article on p-group says "For instance, of the 49 910 529 484 different groups of order at most 2000, 49 487 365 422, or just over 99%, are 2-groups of order 1024 (Besche, Eick & O'Brien 2002)." Am I missing something here?--agr (talk) 15:59, 5 August 2010 (UTC)


 * 423,164,062 is the number of groups of the given type (order at most 2000 and not 1024) in the library. Together with the 49,487,365,422 groups of order 1024, this makes 49,910,529,484 groups of order at most 2000, as the other article says. --Zundark (talk) 16:16, 5 August 2010 (UTC)
 * Thanks for the clarification, but I do find the language used unnecessarily confusing. It might be clearer to say: "those of order at most 2000, except for order 1024 (423 164 062 groups, the excluded groups of order 1024 comprise an additional 49 487 365 422 nonisomorphic 2-groups.)"--agr (talk) 17:22, 5 August 2010 (UTC)

Pauli group - contradiction?
Under groups of order 16, there is a group listed as "the group generated by the Pauli matrices". However on our article Pauli group, this group is stated to be isomorphic to the generalised quaternion group of order 16, which is listed separately on the page. This is clearly contradictory - the two groups are shown to have different cycle graphs. Further, this would give just eight isomorphism classes of non-Abelian groups of order 16, meaning we're missing one. (This would possibly be the group with GAP ID 3 or 13). Kidburla (talk) 16:15, 8 December 2010 (UTC)

Ambiguous link to Marshal Hall
In the citation for Marshal Hall's book on groups of order dividing 64, the link goes to http://en.wikipedia.org/wiki/Marshall_Hall which is a disambiguation page. I believe the correct link should be to http://en.wikipedia.org/wiki/Marshall_Hall_(mathematician) but I can't figure out how to make the citation point there. Somebody who is expert at Wiki please fix it. 198.144.192.45 (talk) 08:20, 27 November 2011 (UTC) Twitter.Com/CalRobert (Robert Maas)


 * I've fixed it. --Zundark (talk) 09:20, 27 November 2011 (UTC)


 * Ah, thanks. 198.144.192.45 (talk) 19:35, 6 January 2013 (UTC) Twitter.Com/CalRobert (Robert Maas)

Why stop at 16?
Is this simply because we've managed to find sources for all the groups of orders up to 16, but not for 18? Or is there some other reason?

Obviously we can trivially extend it to 17. By the fundamental theorem of finite abelian groups we can extend the abelian section as far as we like. Of course, we would need to find a source for the non-abelian groups of orders 18 and 20 if we're going to extend it this far. (It would be nice if we could run a program to generate all the possible group tables for these orders and publish the results here. But that would be OR.)

But 21 is interesting as it's the smallest odd order of a non-abelian group. So it would be nice if we can extend it at least as far as this. — Smjg (talk) 17:49, 1 January 2014 (UTC)

It's not hard to give the groups for 18, 20, and 21. All have prototypes in still smaller groups. But perhaps they are really not that interesting. 21, as a non-abelian group of odd order, might just qualify. Erbach (talk) 18:45, 4 May 2014 (UTC)


 * I'd like to extend this list to order 24, not so many more, and useful since that covers tetrahedral group symmetry, and a few more crystalographic dihedral groups. Tom Ruen (talk) 19:41, 16 December 2014 (UTC)

Notation needs explanation
The second column in the table has been introduced with notation that seems obscure (Goi), and explanation in the article seems appropriate. —Quondum 22:15, 16 December 2014 (UTC)


 * I added a sentence to the Glossary. The naming is used in this paper with cycle graphics up to order 36. I've seen other indexing systems used, so if I (or someone) find other sources with different indexing, we can can cross-reference them also. Tom Ruen (talk) 22:40, 16 December 2014 (UTC)


 * I'm not convinced that indexing to an individual reference's (or in this case actually the small groups library's) indexing scheme is warranted in this table, since this article is not about the small groups library; it only uses it as a reference (though I'm not specifically opposing it). Also, I notice the two indices have been swapped on the non-abelian groups of order 18 and higher. —Quondum 23:27, 16 December 2014 (UTC)


 * Agreed it is excessive and I don't know what might be warrented in the long term, but given my lack of familiarity, and the ease of errors, for the moment, I find it necessary to help me, at least until all the table entries are filled, and like the cycle graphs completed. Tom Ruen (talk) 00:47, 17 December 2014 (UTC)


 * Makes sense. Happy editing.  —Quondum 06:54, 17 December 2014 (UTC)

External links modified
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 * Added tag to http://groupprops.subwiki.org/wiki/Direct_product_of_Z3_and_Z3
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 * Added tag to http://groupprops.subwiki.org/wiki/Direct_product_of_Z12_and_Z2
 * Added archive https://web.archive.org/web/20060923012610/http://math.sun.ac.za/~wild/Marcel%20Wild%20-%20Home%20Page_files/Groups16AMM.pdf to http://math.sun.ac.za/~wild/Marcel%20Wild%20-%20Home%20Page_files/Groups16AMM.pdf
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Low order
The first number given is 0. It is not clear if this is for n=1 or n=0.
 * This is in the paragraph called "Counts". — Preceding unsigned comment added by 2A00:23C0:7C80:8401:E46B:CCC4:EA04:5A30 (talk) 15:06, 18 October 2018 (UTC)
 * Fixed. —David Eppstein (talk) 16:20, 18 October 2018 (UTC)

Symbol
What does ⋊ mean? 31.48.245.78 (talk) 13:19, 19 February 2020 (UTC)
 * Semidirect product. Double sharp (talk) 18:01, 19 February 2020 (UTC)

Small Groups Library
This section says that the smallest order for which the Small Groups Library does not have any information is 2048. That matches neither my experience (OR) nor the GAP documentation.

Both OR and the documentation say that order 1024 is not included.

What is the reason for the discrepancy? Is there something that I'm missing here?

Mstemper (talk) 21:04, 26 June 2020 (UTC)

It has been two months since I asked about the reason that this page currently states that order 2048 is the smallest order missing from the Small Groups Library. In that time, I have received no answer. I assume that this means that this statement is incorrect.

Accordingly, if nobody objects by 2020-09-15, I will fix the two incorrect statements.

Mstemper (talk) 19:54, 5 September 2020 (UTC)

What are Goi in the tables?
What are Goi in the tables, and would someone add a paragraph in the article to explain? Thanks. --Rockyunited (talk) 20:45, 27 December 2020 (UTC)

Number of groups of order 1024
This page (2021) states that there are 49487367289 groups of order 1024. Should the number 49487365422 in the section "Small Groups Library" be corrected? See also this Math Stack Exchange question. 129.104.240.174 (talk) 00:26, 28 August 2023 (UTC)


 * Yes, per comment at Talk:1024_(number). Barnards.tar.gz (talk) 15:20, 28 August 2023 (UTC)

Possible error in semidirect Z4 x Z4
For the semidirect product $$ \mathbb{Z}_4 \rtimes \mathbb{Z}_4 $$ table lists 3 subgroups $$ \mathbb{Z}^{2}_2 \times \mathbb{Z}_2 $$, which seems to be wrong, and instead there should be 3 subgroups $$ \mathbb{Z}_4 \times \mathbb{Z}_2  $$, as for example in this table. Can someone check that? Serpens 2 (talk) 16:27, 29 January 2024 (UTC)


 * I also see the similar issue with $$ Q_8 \times \mathbb{Z}_2 $$ Serpens 2 (talk) 17:42, 29 January 2024 (UTC)
 * and with Iwasawa modular group Serpens 2 (talk) 10:03, 30 January 2024 (UTC)