[[Isomorphism theorems]]
# Group isomorphism theorems

The [[isomorphism theorems]] for [[Module|modules]] are expressed as follows

## First isomorphism theorem

Let $\varphi : G \to H$ be a [[module homomorphism]].
Then the [[Quotient module|quotient]] by the [[Kernel of a module 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 $A,B \leq G$. Then #m/thm/group 
$$
\begin{align*}
\frac{A+B}{B} \cong \frac{A}{A \cap B}
\end{align*}
$$

## Third isomorphism theorem

Let $A \leq B \leq G$.
Then $B / A \leq G / A$ and #m/thm/group 
$$
\begin{align*}
\frac{G / A}{B / A} \cong \frac{G}{B}
\end{align*}
$$

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