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Permutation isomorphic

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Please see the attached file. I need a detailed proof of this to study please.

Permutation groups G1 and G2 acting on the sets S1 and S2 are called permutation isomorphic if there
exists an isomorphism  : G1 ! G2 and a bijection  : S1 ! S2 such that (x)(s) = (xs) 8 x 2 G1 and
s 2 S1. In other words, the following diagram commutes:
S1
x 􀀀! S1
 # # 
S2
􀀀! x S2
De ne two group actions of a group G on itself as follows:
(i) the action of x 2 G is left multiplication by x;
(ii) the action of x 2 G is right multiplication by x􀀀1
Show that the two actions are permutation isomorphic.
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Permutation isomorphic is examined.

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Let theta : G --> G be the identity map, and let phi : G --> G be defined by phi (g) = g^{-1};

then the top ...

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