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## Names of finite groups

Question: If you have a finite group, how do you name it?

If, for whatever reason, you have to list all subgroups of $GL_2({\mathbb F}_5)$ up to isomorphism in a paper, you are likely to write something along the lines of

$$C_1, C_2, C_2, C_3, C_{2,2}, C_4, C_5, C_6, S_3, Q_8, C_8, C_{2,4}, D_4,$$ $$C_{10}, D_5, D_6, C_{12}, C_3\rtimes C_4, C_{2,4}\rtimes C_2, OMC_{16}, C_{4,4},$$ $$C_{20}, D_{10}, G_{20}, C_5\rtimes C_4, SL_2(F_3), C_4\times S_3, C_3\rtimes C_8, C_{24},$$ $$Q_8\rtimes C_4, C_2\times G_{20}, C_2\times G_{20}, C_4\times D_5, (C_{2,4}\rtimes C_2)\rtimes C_3, C_3\rtimes OMC_{16},$$ $$C_4\times G_{20}, C_2.A_5, SL_2(F_3)\rtimes C_4, (C_2.A_5)\rtimes C_2, GL_2(F_5).$$

Computer algebra packages tend to produce a human-unfriendly output of generators and relations or generating permutations in $S_n$. How do you convert from one to the other and decide how to name complicated groups? I am looking for standard names, standard constructions, conventions and notations. For me a good notation is informative, human friendly, short and is generally as close as possible to what you would use in a paper. I am also looking for any kind of canonical conventions: e.g. $(C_5\times C_5)\rtimes C_4$ or $(C_5\rtimes C_4)\times C_5$?

(The reason I am asking is that I seem to have to work with funny groups all the time recently. I have a Magma function for personal use that analyzes and names finite groups; e.g. it produces the list above for $GL_2({\mathbb F}_5)$, and I personally find this really useful.

Currently it knows various standard groups: cyclic, abelian, dihedral, alternating, symmetric, special $p$-groups (semi-dihedral, generalized quaternion, "other maximal cyclic", Heisenberg), simple groups, linear groups (SL, GL, O, SP) and eventually their projective versions; it tries to recognize direct, semidirect (and eventually wreath) products if the group is not too large, and reverts to chief series if everything else fails.

Recently sufficiently many people asked me to share the code that I'll make it public domain. But before that I'd very much like to get suggestions from the MO community how to make it as useful as possible for most people.)

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$G_{20}$? I'd call it $F_{20}$ as it is the Frobenius group of order $20$. So good luck with finding names! – Someone Dec 6 2010 at 12:48
Jonathan, I thought the notation for the dihedral groups is standard: group theorists write the dihedral group of order 2n as D_{2n} and everyone else (?) writes the group as D_n. – KConrad Dec 6 2010 at 13:34
@KConrad: I suppose I'll have an option IAmAGroupTheorist:boolean in my code to deal with dihedral groups then. @Jonathan: My OMC16 comes from the last paragraph of en.wikipedia.org/wiki/Quasidihedral_group, I don't know if that's the same as a modular group. What is the modular group of order 16 (as generators and relations or whatever)? – Tim Dokchitser Dec 6 2010 at 13:50
Unfortunately, we all still publish paper in Dead Tree. If you use a more modern graphical user interface, namely Light Emitting Diode, then you can usually provide greater functionality to your readers: allow them to click or double click or right click or whatever on each name for more information on it. (Such hyperlinking is also available in Dead Tree, of course, in the form of footnotes, endnotes, appendices, and references.) – Theo Johnson-Freyd Dec 6 2010 at 16:23
Have you looked into what the GAP function StructureDescription currently does? Details can be found in the manual: gap-system.org/Manuals/doc/htm/ref/… – ndkrempel Dec 6 2010 at 19:43

It is difficult to come up with a consistent notation for all groups of a certain order since their construction is somewhat chaotic. We might be able to describe all the groups of order $p^3$ or $p^4$ but what about all groups of order $p^6$? Or order $p^4q^2$?

The software package GAP (http://www.gap-system.org/) has a catalogue of all groups of order up to 2000 or so and so I've sometimes referred to groups by their catalogue number, for example, SmallGroup(96, 33) refers to a particular group in that library. (As does SmallGroup(512, 1000000)!)

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I agree, one might use the small group database if there is really no other choice, but generally this is not very human friendly. E.g. SmallGroup(96,33) is $C_3\rtimes D_{16}$, which may not describe it uniquely but at least it says more. – Tim Dokchitser Dec 6 2010 at 13:56
In GAP you can try StructureDescription: gap> StructureDescription(SmallGroup(96,33)); "(C3 x D16) : C2" – Primoz Dec 7 2010 at 20:11

There is a useful convention to decorate some of the groups with an index which is the smallest $n$ for which the group can act transitively on $n$ points, i.e. embeds in $S_n$ as a transitive subgroup. The notation for $S_n, A_n, D_n, C_n$, your $Q_8$ and for example Mathieu groups $M_{11}, M_{12}, M_{22}$ (although not other sporadic simple groups) follow this pattern.

Of course, there is also another convention to use the size of the group instead...

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... and, as mathematicians would put it, "This problem is a natural generalization of the dihedral case" – Tim Dokchitser Dec 8 2010 at 0:19
It's not clear to me that n is the smallest index k for which D_n embeds into S_k. – Qiaochu Yuan Dec 8 2010 at 0:43
That's right, $D_{15}$ can act faithfully on $3+5=8$ points. Smallest $n$ for which $G$ is a transitive subgroup of $S_n$? – Tim Dokchitser Dec 8 2010 at 0:48

For transitive permutation groups the first paper in Journal of Computation & Mathematics by Conway, Hulpke, & McKay lists the smaller degrees with "respectable names".

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 Thank you, I'll have a look! – Tim Dokchitser Apr 24 2011 at 19:59