# Properties of a group

### Some properties of a group $G$

Let $K\lhd G$, $K\subset H$ be a subgroup of $G$.

##### 1

Map
\begin{align}
\begin{array}{ccc}
\{H\,|\,K\subset H:\text{subgroup of } G\}
&\rightarrow
&\{\bar{H}\,|\bar{H}:\text{subgroup of } G/K\}\\
H&\mapsto&H/K
\end{array}
\end{align}
is bijective.

• Proof. Let $\bar{H}= \{aK\,|\,a\in A\}$. Then $\displaystyle\bigcup_{a\in A}aK\in G$ and $\displaystyle\bar{H}= \{aK\,|\,a\in A\}\mapsto \bigcup_{a\in A}aK$ is an inverse.
##### 2

\begin{align}
(K\subset)H\lhd G
\Leftrightarrow
H/K\lhd G/K
\end{align}

• Proof. Easily shown from the definition.
##### 3

If $(K\subset)H\lhd G$
\begin{align}
G/H \cong (G/K)\bigl/(H/K)
\end{align}

• Proof. Kernel of the homomorphism \begin{align}\begin{array}{cccc}G&\rightarrow&G/K&\rightarrow&(G/K)\bigl/(H/K)\\g&\mapsto&gK&\mapsto& gK(H/K)\\&&&\end{array}\end{align} is $H$.

Now, let $f:G\rightarrow G^\prime$ be epimorphism. Then we can get following generalised results(*1) by isomorphism $G/\mathrm{Ker\,}f\ni g\mathrm{Ker\,}f\mapsto f(g)\in G^\prime$.

##### 1'

Map
\begin{align}
\begin{array}{ccc}
\{H\,|\,\mathrm{Ker\,}f\subset H:\text{subgroup of } G\}
&\rightarrow
&\{H^\prime\,|H^\prime:\text{subgroup of } G^\prime\}\\
H&\mapsto&f(H)
\end{array}
\end{align}
is bijective.

##### 2'

\begin{align}
(\mathrm{Ker\,}f\subset)H\lhd G
\Leftrightarrow
f(H)\lhd G^\prime
\end{align}

##### 3'

If $(\mathrm{Ker\,}f\subset)H\lhd G$
\begin{align}
G/H \cong G^\prime/f(H)
\end{align}

*1:Consider the case $f:G\ni g\mapsto gK\in G/K$.