(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

What is the center and the commutant of GL(n = 2, Z_p)?

2. Relevant equations

The center of a group G: z(G) = {x in G| xy = yx for all y in G}

Commutant of a group G is the set of all [itex] xyx^{-1}y^{-1} [/itex] where all x, y are in G.

3. The attempt at a solution

I tried some calculations with p = 2 first to see if some pattern emerged, but gave up along the way multiplying matrices. My idea is that since I can get the order of GL(n = 2, Z_p) for any p, then I could try to construct an isomorphism between this group and the dihedral group that has the same order. Since the center and commutant of this dihedral group is much easier to get, then I will have obtained the center and commutant of the GL group. Is this right? I gave a try to p=2 that gives order 6 to GL and was able to construct an isomorphism successfully with D_3. My problem now is trying to prove that I can show the isomorphism between any GL(n = 2, Z_p) and the corresponding dihedral group with the same order. Thanks for any help.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Center and Commutant: GL(n = 2, Z_p)

**Physics Forums | Science Articles, Homework Help, Discussion**