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Consider the group of permutation matrices $G =\{I_3, P_1, P_2, P_3, P_4, P_5\}$ For $n=3$ the permutation matrices are $I_3$ and the five matrices are:

\begin{equation*} P_1 = [1,0,0;0,0,1;0,1,0] \\ P_2 = [0,1,0;1,0,0;0,0,1] \\ P_3 = [0,1,0;0,0,1;1,0,0] \\ P_4 = [0,0,1;0,1,0;1,0,0] \\ P_5 = [0,0,1;1,0,0;0,1,0] \end{equation*}

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 $f_a:G\to G$ 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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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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