First group isomorphism theorem

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Note:

• Overview of the group isomorphism theorems - all 3 theorems in one place
• Overview of the isomorphism theorems - the first, second and third are pretty much the same just differing by what objects they apply to

First isomorphism theorem
$\begin{xy}\xymatrix{A \ar[r]^\varphi \ar[d]_{\pi} & B \\ A/\text{Ker}(\varphi) \ar@{.>}[r]^-{\theta}& \text{Im}(\varphi) \ar@{^{(}->}[u]^i }\end{xy}$ Where $\theta$ is an isomorphism.
Properties
something

Statement

Let $(G,*)$ and $(H,*)$ be groups. Let $\varphi:G\rightarrow H$ be a group homomorphism, then:

• $G/\text{Ker}(\varphi)\cong\text{Im}(\varphi)$
  • Explicitly we may state this as: there exists a group isomorphism between $G/\text{Ker}(\varphi)$ and $\text{Im}(\varphi)$.

Note: the special case of $\varphi$ being surjective, then $\text{Im}(\varphi)=H$, so we see $G/\text{Ker}(\varphi)\cong H$

Useful corollaries

1. An injective group homomorphism means the group is isomorphic to its image
   • If $\varphi:A\rightarrow B$ is an injective group homomorphism then $A\cong \text{Im}(\varphi)$
2. A surjective group homomorphism means the target is isomorphic to the quotient of the domain and the kernel
   • If $\varphi:A\rightarrow B$ is a surjective group homomorphism then $A/\text{Ker}(\varphi)\cong B$

Proof

• See Notes:Proof of the first group isomorphism theorem