TITLE: Show that $H$ is not normal in $A_4$?
QUESTION [0 upvotes]: Let $H=\{(1), (1,2)(3,4)\}$ in $A_4$. Show $H$ is not Normal in $A_4$ using the definition of normal. 
So I know that $A_4$ is the alternating group on $4$ letters, but I don't understand what that means. Can someone please explain? I also know that for a subgroup $H$ of $G$ to be normal, its left and right cosets must coincide. ie: $aH=Ha$. From here, how do I use this definition to prove that $A_4$ is not normal?
My plan was this: 
the left cosets are:
$1+H= \{(2),(2,3),(4,5)\}$ $\leftarrow$ Can you have $5$ in $A_4$, or would it be $0$ or $1$???
$2+H= \{(3),(3,4)\}$

$3+H= \{(4)\}$,
I think my main issue is that I do not understand what it means for something to be an "alternating group on 4 letters". Also, am I on the right track for trying to see if the cosets coincide?

REPLY [1 votes]: Hint:
$$(1\;2\;3)\color{red}{(1\;2)(3\;4)}(1\;2\;3)^{-1}=(123)\color{red}{(12)(34)}(132)=(14)(23)\notin H$$
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