Normal Subgroups and Quotient Groups Review

# Normal Subgroups and Quotient Groups Review

We will now review some of the recent material regarding normal subgroups and quotient groups.

• On the Normal Subgroups we said that if $(G, *)$ is a group and $(H, *)$ is a subgroup, then $(H, *)$ is said to be a Normal Subgroup if for all $g \in G$ we have that:
(1)
\begin{align} \quad gH = Hg \end{align}
• That is, the left coset of $H$ with representative $g$ is equal to the right coset of $H$ with representative $g$ for every $g \in G$.
• We noted that if $(G, *)$ is an abelian group then every subgroup of $(H, *)$ is a normal subgroup.
Equivalent Statements
(a) $(H, *)$ is a subgroup of $(G, *)$.
(b) $gHg^{-1} \subseteq H$ for all $g \in G$.
(c) $gHg^{-1} = H$ for all $g \in G$.
• On the Quotient Groups page we said that if $(G, *)$ is a group and $(H, *)$ is a normal subgroup then the corresponding Quotient Group is the set $G / H = \{ gH : g \in G \}$ of left cosets of $H$ with the operation $\cdot$ defined for all $g_1H, g_2H \in G / H$ by:
(2)
\begin{align} \quad (g_1H) \cdot (g_2H) = (g_1 * g_2)H \end{align}
• We proved that the order of $G / H$ is given by:
(3)
\begin{align} \mid G / H \mid = \frac{\mid G \mid}{\mid H \mid} \end{align}