[[Isomorphism theorems]]
# Group isomorphism theorems

The [[isomorphism theorems]] for [[Group|groups]] are expressed as follows

## First isomorphism theorem

Let $\varphi : G \to H$ be a [[Group homomorphism]].
Then the [[Quotient group|quotient]] by the [[Kernel of a group homomorphism|kernel]] is isomorphic to the image: #m/thm/group 
$$
\begin{align*}
\frac{G}{\ker \varphi} \cong \im \varphi \leq H
\end{align*}
$$

## Second isomorphism theorem

Let $A,B \trianglelefteq G$. Then #m/thm/group 
$$
\begin{align*}
\frac{AB}{B} \cong \frac{A}{A \cap B}
\end{align*}
$$

## Third isomorphism theorem

Let $A, B \trianglelefteq G$ be [[Normal subgroup|normal subgroups]] so that $A \leq B$.
Then $B / A \trianglelefteq G / A$ and #m/thm/group 
$$
\begin{align*}
\frac{G / A}{B / A} \cong \frac{G}{B}
\end{align*}
$$

#
---
#state/tidy | #lang/en | #SemBr