\(
\newcommand{\cat}[1] {\mathrm{#1}}
\newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})}
\newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}}
\newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}}
\newcommand{\betaReduction}[0] {\rightarrow_{\beta}}
\newcommand{\betaEq}[0] {=_{\beta}}
\newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}}
\newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}}
\newcommand{\groupMul}[1] { \cdot_{\small{#1}}}
\newcommand{\groupAdd}[1] { +_{\small{#1}}}
\newcommand{\inv}[1] {#1^{-1} }
\newcommand{\bm}[1] { \boldsymbol{#1} }
\require{physics}
\require{ams}
\require{mathtools}
\)
Math and science::Algebra::Aluffi
Groups. Monomorphism equivalents.
\( \text{monomorphism} \iff \text{kernel} = \{e\} \iff
\text{set-injective} \)
Let \( G \) and \( H \) be groups, and \( \varphi : G \to H \) be a group
homomorphism.
The following are equivalent:
- \( \varphi \) is a monomorphism.
- \( \ker \varphi = \{ e_G\} \).
- \( \varphi \) is injective as a set-function.
The implication from 2) to 3) reminds me of Lagrange's theorem.
[Can you remember the proofs?]
