5
$\begingroup$

I'm really looking for a "cute" way of showing that $SL_2(\mathbb{F}_5)$ is a double cover of $A_5$. The sort of action I am looking for is something like the action of $GL_2(\mathbb{F}_3)$ on $\mathbb{P}^1(\mathbb{F}_3)$, which shows $GL_2(\mathbb{F}_3)$ is a double cover of $S_4$. Now that's cute.

$\endgroup$
1
$\begingroup$

You can ask this type of questions to GAP:

GAP4, Version: 4.4.12 of 17-Dec-2008, x86_64-unknown-linux-gnu-gcc
gap> g := SL(2,5);;
gap> 5 in List(ConjugacyClassesSubgroups(g), c -> Index(g, Representative(c)));
true
gap> 

There is in fact a unique conjugacy class of subgroups of index 5, isomorphic to SL(2,3).

$\endgroup$
  • $\begingroup$ Smart! I was looking at conjugacy classes of elements but didn't think of looking at the subgroup lattice. There's also a conjugacy class of subgroups isomorphic to the quaternion group $Q_8$. $\endgroup$ – Joe Mar 29 '12 at 18:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.