Criteria for a Subgroup to be Normal

Criteria for a Subgroup to be Normal

Recall from the Normal Subgroups page that if $(G, *)$ is a group and $(H, *)$ is a subgroup then $(H, *)$ is said to be a normal subgroup if $gH = Hg$ for all $g \in G$, that is, the left and right cosets of $H$ with representative $g$ are equal for all $g \in G$.

We will now look at some criteria for when a subgroup of a group will be normal.

Theorem 1: Let $(G, *)$ be a group and let $(H, *)$ be a subgroup. Then the following statements are equivalent:
a) $(H, *)$ is a normal subgroup of $(G, *)$.
b) $gHg^{-1} \subseteq H$ for all $g \in G$.
c) $gHg^{-1} = H$ for all $g \in G$.
  • Proof of $a) \implies b)$ Suppose that $(H, *)$ is a normal subgroup of $(G, *)$. Let $x \in H$. Then since $H$ is normal we have that $gH = Hg$ for all $g \in G$ and so $gx = xg$ for all $g \in G$. So $x = gxg^{-1}$. But $gHg^{-1} = \{ gxg^{-1} : x \in H \}$ and $x \in H$ which shows that for all $g \in G$ we have that:
(1)
\begin{align} \quad gHg^{-1} \subseteq H \end{align}
  • Proof of $b) \implies c)$ Suppose that $gHg^{-1} \subseteq H$ for all $g \in G$. Let $x \in H$ and let $g \in G$. Then:
(2)
\begin{align} \quad g^{-1}xg = g^{-1}x(g^{-1})^{-1} \in g^{-1}N(g^{-1})^{-1} \end{align}
  • So there exists an $x' \in H$ such that $g^{-1}x(g^{-1}) = x'$, i.e., $x = gx'g^{-1}$ for all $g \in G$. But then for all $g \in G$, $x \in gHg^{-1}$ i.e., for all $g \in G$ we have $gHg^{-1} \supseteq H$ which shows for all $g \in G$ that:
(3)
\begin{align} \quad gHg^{-1} = H \end{align}
  • Proof of $c) \implies a)$ Suppose that $gHg^{-1} = H$ for all $g \in G$. Let $x \in H$. Then for all $g \in G$ there exists and element $x' \in H$ such that $x = gx'g^{-1}$. But then $xg = gx'$ for all $g \in G$ which shows that $gH = Hg$ for all $g \in G$. Thus $(H, *)$ is a normal subgroup of $(G, *)$. $\blacksquare$
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