If $G=S_4, N=\{e;(1,2)(3,4);(1,3)(2,4);(1,4)(2,3)\}$ show that $N\lhd G$

Question

Let $G=S_4, N=\{e;(1,2)(3,4);(1,3)(2,4);(1,4)(2,3)\}$

I need to prove that $N\lhd G$

Attempt:

$N\lhd G \iff gng^{-1}\in N$

and it is in $N$ for all $g\in G, n\in N$

That subgroup is called the Klein 4-group. – Rachid Atmai Dec 15 '19 at 01:23 — Rachid Atmai, Dec 15 '19 at 01:23

score 1 · Accepted Answer · answered May 19 '15 at 16:49

You need to check two things.

$N$ is a subgroup of $G$. For this, you may check that every element is its own inverse, and that the product of $(12)(34)$ and $(13)(24)$ is $(14)(23)$, the rest follows by symmetry. (Or prove that if $a, b, c, d$ are four distinct elements, then $(ab)(cd)$ times $(ac)(bd)$ is $(ad)(bc)$.)
$N$ is normal. For this, you may use the fact that the conjugate of a permutation $n$ has the same cycle structure of $n$. Then note that the non-identity elements of $N$ are all the elements of $G$ that are the product of two distinct 2-cycles.

1 Answers1