0
$\begingroup$

Let $H$ ={$e, (1, 2) (3, 4)$} and $K$ ={$e, (1, 2) (3, 4), (1, 3) (2, 4), (1, 4) (2, 3)$} be subgroups of $S_4$, where $e$ denotes the identity element of $S_4$. Then

  1. $H$ and $K$ are normal subgroups of $S_4 $.
  2. $H$ is normal in $K$ and $K$ is normal in $A_4$.
  3. $H$ is normal in $A_4$ but not normal in $S_4$.
  4. $K$ is normal in $S_4$, but $H$ is not.

How should I able to solve this problem. Can anyone help me please.

  • 0
    How does the cycle structure of an element $x$ relate to the cycle structure of $axa^{-1}$?2019-02-08

1 Answers 1

1

Hints:

$\begin{align*}(a)&\;(123)\left[(12)(34)\right](132)=(14)(23)\notin H\\ (b)&\;\text{For any group}\,G\,\,\text{and any subgroup}\,H\leq G\,\,,\,[G:H]=2\Longrightarrow H\triangleleft G\\ (c)&\;\text{Two permutations are conjugate in }\,S_n\,\text{ iff they have the same cyclic decomposition}\end{align*}$

The above answers all you need: (1) and (3) are false.