Proving Non-Normality: A_4 in S_4

[fk378]

Homework Statement

Prove by an example, that we can find 3 groups E c F c G, where E is normal in F, F is normal in G, but E is not normal in G.
(c denotes "contained in")

Consider:
G=A_4
F={Id, (12)(34), (13)(24), (14)(23)} and
E={Id, (12)(34)}

The Attempt at a Solution

I want to use the fact that A_4 is the kernel of S_4 so automatically it is normal in S_4. But how is F related to S_4?

 

[morphism]

What do you mean "A_4 is the kernel of S_4"?

Forget about S_4. Consider the examples you have been given. E is a subgroup of F which in turn is a subgroup of G. Is E normal in F? In G? Is F normal in G?