**normal subgroup**of $G$ if $aH=Ha$ for every $a\in G$, and we write $H\u22b4G$

Normal Subgroup

A subgroup $H$ of $G$ is called a **normal subgroup** of $G$ if $aH=Ha$ for every $a\in G$, and we write $H\u22b4G$

Normal Subgroup Test

A subgroup $H$ of $G$ is normal iff $xH{x}^{-1}\subseteq H$ for every $x\in G$

Quotient Group

Suppose that $G$ is a group and that $H\u22b4G$. Then we define $$G/H:=\{aH:a\in G\}$$

Some may call a quotient group a factor group

Solvable

Show that ${S}_{3}$ is solvable

The Symmetric Group for n Greater Than or Equal to 5 Is not Solvable

${S}_{n}$ for $n\ge 5$ is not solvable