Group Homomorphism • Lutalli

DEF-GH. Group Homomorphism.

A group homomorphism between two groups $(G,*)$ and $(H,\diamond)$ is a function $f:G\to H$ such that for all $a$, $b\in G$,
\[f(a*b) = f(a)\diamond f(b).\]

DEF-GH-KER. Kernel.

The kernel of a group homomorphism $f:G\to H$ is
\[\ker f = f^{-1}[\{i\}],\]
where $i$ is the neutral element of $H$.

PROP-GH-A.

Let $f:G\to H$ be a group homomorphism, $e$ be the neutral element of $G$ and $i$ be the neutral element of $H$.

\[f(e)=i.\]

\[f(a^{-1})=f(a)^{-1}\]
for every $a\in G$.

PROP-GH-KER.

Let $f:G\to H$ be a group homomorphism. $\ker f$ is a normal subgroup of $G$.

PROP-GH-IM.

Let $f:G\to H$ be a group homomorphism. $\im f$ is a subgroup of $H$.

PROP-GH-K.

Let $f:G\to H$ be a group homomorphism. For any $a$, $b\in G$,
\[f(a) = f(b) \enspace\lrimp\enspace ab^{-1}\in\ker f.\]

PROP-GH-MON. Corollary of PROP-GH-K

Let $f:G\to H$ be a group homomorphism and $e$ be the neutral element of $G$. $f$ is a monomorphism if and only if
\[\ker f = \{e\}.\]