Hello, I was wondering if I could get some help with this question.
Let G be the subgroup of generated by (1 2),(3 4) and (1 3)(2 4). Let H be the subgroup of S_4 generated by (1 2) and (3 4). Show that H is a normal subgroup of G. Show that H has order 4 and G/H has order 2. Deduce that G has order 8.
I was wondering if there is a quick way to show that G has order 4 without doing it explicitly? I was also wondering how to show that G/H has order 4. I guess that Lagrange's Theorem proves the last part. Any help with this would be appreciated.
To see this you need to know that a group is normal if every left coset is a right coset. And if a subgroup has index two this is easy to see since we have that the set of left cosets looks like and the set of right cosets looks like and since and it followst hat .
I understand that [G:H] = | G / H | if H is normal to G and I know that |G| = 8.
But how do I show that [G:H] = 2 only knowing that H has order 4. The question does not ask me to work out |G| but to deduce it from the fact that [G:H] = 2, but how do I show this without showing |G| = 8 first.
I am trying to show that there must only be 2 distinct cosets of H in G but Im not getting very far