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
[ilmath]\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}[/ilmath] Where [ilmath]\theta[/ilmath] is an isomorphism.
Properties
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Statement

Let [ilmath](G,*)[/ilmath] and [ilmath](H,*)[/ilmath] be groups. Let [ilmath]\varphi:G\rightarrow H[/ilmath] be a group homomorphism, then:

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

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

Useful corollaries

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

Proof

• See Notes:Proof of the first group isomorphism theorem