Names of finite groups - MathOverflow

Names of finite groups

2010-12-06T12:08:39Z

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.)

Answer by Ken W. Smith for Names of finite groups

2010-12-06T13:47:12Z

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)!)

Answer by M.Z. for Names of finite groups

2010-12-08T00:17:23Z

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...

Answer by JMcKay for Names of finite groups

2011-04-24T12:47:15Z

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