Group monomorphism

Let $𝜑 : 𝐺 \to 𝐻$ be a group homomorphism. The following statements are equivalent: #m/thm/group

$𝜑$ is injective;
$𝜑$ is monic;
$k e r 𝜑 = 1$

Proof

$𝜑$ is injective iff $𝜑 (𝑎) = 𝜑 (𝑏) ⟹ 𝑎 = 𝑏$ iff $𝜑 (𝑎 𝑏 - 1) = 𝑒 ⟹ 𝑎 𝑏 - 1 = 𝑒$ iff $k e r 𝜑 = {𝑒}$ . Clearly injective $𝜑$ implies monic $𝜑$ . Now let $𝜑 : 𝐺 ↣ 𝐻$ be a monomorphism. Let $𝜄 : k e r 𝜑 ↪ 𝐺$ be the canonical injection and $𝛼 𝑇 : k e r 𝜑 \to 𝐺 : 𝑥 \mapsto 𝑒$ be the trivial homomorphism. Since $𝜑 𝜄 = 𝜑 𝛼 𝑇$ , $𝜄 = 𝛼 𝑇$ and hence $k e r 𝜑 = {𝑒}$ .

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