Subgroups of GL_2 over a finite field - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T21:39:02Z http://mathoverflow.net/feeds/question/97052 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97052/subgroups-of-gl-2-over-a-finite-field Subgroups of GL_2 over a finite field green jeans 2012-05-15T21:04:48Z 2012-05-16T15:09:28Z <p>I've come across the phrase "by the classification of subgroups of \$GL_2(F_q)\$" in multiple papers, but never with a reference. Here \$F_q\$ is a finite field of size \$q\$. Does anyone know a good reference for this (ideally for someone who is not a group theorist)?</p> http://mathoverflow.net/questions/97052/subgroups-of-gl-2-over-a-finite-field/97054#97054 Answer by Barinder Banwait for Subgroups of GL_2 over a finite field Barinder Banwait 2012-05-15T21:13:41Z 2012-05-15T21:13:41Z <p>At the text-book level, take a look at Lang's Algebra, Chapter XVIII, Section 12, in my version is on page 712. Seems to be pretty thorough. </p> <p>There is also section 2 of Serre's 1972 ``Propriétés galoisiennes des points d'ordre fini des courbes elliptiques''. </p> http://mathoverflow.net/questions/97052/subgroups-of-gl-2-over-a-finite-field/97096#97096 Answer by Dima Pasechnik for Subgroups of GL_2 over a finite field Dima Pasechnik 2012-05-16T08:23:15Z 2012-05-16T08:23:15Z <p>This classification was well-known already in the beginning of XXth century; perhaps this explains why people do not bother to give a reference (the result was due to E.H.Moore and, independently, Wiman). </p> <p>Many years ago I read the paper by Howard H. Mitchell, <a href="http://www.ams.org/journals/tran/1911-012-02/S0002-9947-1911-1500887-3/home.html" rel="nofollow">Determination of the ordinary and modular ternary linear groups.</a> Trans. Amer. Math. Soc. 12 (1911), no. 2, 207–242, which addresses the analogous question for \$GL_3(\mathbb{F}_q)\$, and also contains the detailed information on the subgroups of \$GL_2(\mathbb{F}_q)\$. (It's of course not so easy to read, as the terminology was quite different...)</p> http://mathoverflow.net/questions/97052/subgroups-of-gl-2-over-a-finite-field/97129#97129 Answer by Nick Gill for Subgroups of GL_2 over a finite field Nick Gill 2012-05-16T15:09:28Z 2012-05-16T15:09:28Z <p>Dickson is responsible for the classification of subgroups of \$SL_2(\mathbb{F}_q)\$ (and once you've got this the subgroups of \$GL_2(\mathbb{F}_q)\$ are easy). You can find a full proof in Suzuki's "Group Theory Part I". It's Chapter 6, Section 3 of that book. If you email me I'll even send you a copy :-)</p>