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)

D_n

What is D_n 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 × 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 D_n is the dihedral group of order 2n. The isomorphisms given (e.g., D₆ = D₃ × 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 × Z₂

The cycle graph of Q₈ × 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 D_n 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 Z₄²

Shouldn't there be 3 pairs of 4-cycles which share the square element? i.e. take the diagram for Z₄ × 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)

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)