The problem asks to consider the subgroup of S(4) consisting of the four permutations id,(12)(34),(13)(24),(14)(23) and the group of symmetries of a rectangle and find all the isomorphism between these two groups. They already defined one by taking id to e, (12)(34) to σ, (13)(24) toRand (14)(23) to τ (τ=σR).

The hint says "Choose any two elements of order 2; what are the restrictions on where an isomorphism can take them? Having determined where these two elements are sent by the isomorphism, is there any choice for the destinations of the other two elements?

I'm not sure what they mean by restriction. I looked at the multiplication tables of the two and it seemed like as long id was sent to e, there would be an isomorphism between the two groups. So, I came up with 5 more by taking

1. id to e, (12)(34) to σ, (14)(23) toR,and (13)(24) to τ.

2. id to e, (13)(24) to σ, (12)(34) toR,and (14)(23) to τ.

3. id to e, (13)(24) to σ, (14)(23) toR,and (12)(34) to τ.

4. id to e, (14)(23) to σ, (12)(34) toR,and (13)(24) to τ.

5. id to e, (14)(23) to σ, (13)(24) toR,and (12)(34) to τ.

Is this correct? I'm a bit lost with the topic of groups, so any help is appreciated.