Let $A=\{(1), (234), (243) \}$. Determine if $A$ is a normal subgroup of $S_4$.

Question

Let $A=\{(1), (234), (243) \}$. Determine if $A$ is a normal subgroup of $S_4$.

So I have two definitions to determine if a subgroup is normal. If the left and right cosets are equal that is $\alpha A = A\alpha$, where $\alpha \in S_4$. The other one is that $\alpha \beta \alpha^{-1} \in A$ for every $\beta \in A$ and $\alpha \in S_4$.

I don’t really see how I can use either one of these here. $S_4$ has $24$ elements so checking all of them would take quite a while. What can I do here?

Answer 1

Hint:

Recall that a two cycle is self inverse.

Observe that elements of $A$ are even permutations of $S_4$ and that they fix $1$.

So try to find $a\in \{2,3,4\}$ such that $(1a)(234)(1a)^{-1}$ doesn't fix $1$ and you are done using normal subgroup definition.

Answer 2

A normal subgroup is a union of conjugacy classes. If it contains $b$ then it must also contain all the elements conjugate to $b$.

Two permutations are conjugate just when they have the same cycle structure. For $A$ to be normal it would have to contain every permutation with the same cycle structure as $(234)$. Does it?