Timeline for Finite subgroups of GL_n(C)
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| May 16, 2010 at 22:20 | comment | added | Steve D | Perhaps it is interesting to note that both theorems of Blichfeldt mentioned here can be found in his group theory book with Dickson and Miller. The proofs there don't require much character theory IIRC, but are much longer than the ones you can find in a Robinson or Isaacs. The book is available freely (and legally) from Google Books. | |
| May 15, 2010 at 17:12 | comment | added | Jack Schmidt | The specific case of exponent 5 in GL(3,C) follows from two facts: (1) if a group has a faithful representation that is a direct sum of 1-dimensional representations, then it is abelian, and (2) if a group has an irreducible representation of dimension d, then d divides the order of the group. A group of exponent 5 has order a power of 5, so the dimension of its representations are powers of 5. The only power of 5 less than or equal to 3 is 5<sup>0</sup> = 1, and so the only 5-subgroups of GL(3,C) are abelian. The general case is Blichfeldt's theorem 14.1 in Isaacs's Character Theory book. | |
| May 15, 2010 at 17:07 | comment | added | Jack Schmidt | It'll take a bit to write up a simpler proof of Blichfeldt, but the idea of that section of Herzog-Praeger is very standard and effective: (1) if G is a subgroup of GL(M), M a vector space, then M is a G-module, (2) handle the case where M is reducible first (lemma 2.1), (3) handle the case where M is imprimitive second (Blichfeldt), and (4) handle the primitive cases. 1,2,3 are standard and usually "easy", and the groups that survive to 4 usually have a very special structure. Blichfeldt's lemma basically says no primitive group is a nonabelian p-group, which might be familiar for perm reps. | |
| May 15, 2010 at 16:33 | comment | added | Portland | @ Jack, How do show that "For instance, GL(3,C) contains no non-abelian group of exponent 5." Because GL(3,C) contains for instance a non-abelian group of exponent 3 (isomorphic to $(\mathbb{Z} /3 \mathbb{Z} \times \mathbb{Z} /3 \mathbb{Z} ) \rtimes \mathbb{Z} /3 \mathbb{Z}$). | |
| May 15, 2010 at 16:27 | vote | accept | Portland | ||
| May 15, 2010 at 16:09 | comment | added | Portland | Thanks Jack, I knew of that paper, although I had a hard time with Blichfelt lemma (p. 218). I wonder if there is a way around? | |
| May 15, 2010 at 16:00 | history | answered | Jack Schmidt | CC BY-SA 2.5 |