Showing Subgroups of a Permutation Group are Isomorphic

Obraz35

Define two subgroups of S6:
G=[e, (123), (123)(456)]
H=[e, (14), (123)(456)]

Determine whether G and H are isomorphic.

It seems as if they should be since they have the same cardinality and you can certainly map the elements to one another, but I don't know what other factors need to be considered when deciding whether they are isomorphic.

crazyjimbo

H is not a subgroup since (14)(123)(456) = (123456) which is not in H. Are you sure you've written down the correct group?

Obraz35

Oops. I meant G=<(123), (123)(456)> and H=<(14), (123)(456)>.