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Consider the group of permutation matrices G ={I3, P1, P2, P3, P4, P5} For n = 3 the permutation matrices are I3 and the five matrices are: P1 = [1,0,0;0,0,1;0,1,0] P2 = [0,1,0;1,0,0;0,0,1] P3 = [0,1,0;0,0,1;1,0,0] P4 = [0,0,1;0,1,0;1,0,0] P5 = [0,0,1;1,0,0;0,1,0]

Write out the elements of a group of permutations that is isomorphic to G, and exhibit an isomorphism from G to this group!

I think it has to do with Cayley's Theorem. With f_a:G->G defined by f_a(x) = ax for each a that exists in G...

I thought about making a table, but realize I don't know how to since I am dealing with matrices.

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1 Answer 1

HINT: Look at the result of multiplying each of those matrices by the vector $(1,2,3)^T$.

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Oh ok so P1=[132] P2=[213] P3=[231] P4=[321] P5=[312] –  Jessica Oct 22 '10 at 0:43
    
SO now... Isomorphism would be the mapping of G->G' ? –  Jessica Oct 22 '10 at 0:44
    
Well, you have to check that it's an homomorphism. –  Weltschmerz Oct 22 '10 at 1:27
    
Multiplying each of those by the vector did not come out to the isomorphisms... –  Jessica Oct 27 '10 at 20:00
    
check out how i solved this problem... this one i then solved similar –  Jessica Oct 27 '10 at 20:01

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