I more or less copied the following notes from my professor:

Everything about this proof makes perfect sense to me except the part in bold: Why is

a subgroup (isomorphic) of

? It cannot be simply because

acts on the four roots (because for instance any group of any order acts on any set trivially). All I can tell from that is that there is a homomorphism from

into

. So, how do I conclude that

is a subgroup (isomorphic) of

?

Also, I'm a bit confused as to how it can be that

. For each

is completely determined by

and

. Now,

, which seems to imply

. But then

must be one of the four roots. So

, a contradiction. Where did I go wrong, I wonder?

Any help would be much appreciated!