Difference between revisions of "First group isomorphism theorem"

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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

Let $(G,*)$ and $(H,*)$ be groups. Let $\varphi:G\rightarrow H$ be a group homomorphism, then^[1]:

G/Ker(φ)≅Im(φ)
- 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 $G/\text{Ker}(\varphi)\cong H$

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 {{Infobox
 |title=<span style="font-size:0.85em;">First isomorphism theorem</span>
-|above=<span style="font-size:1.3em;">{{M|1= $\begin{xy}\xymatrix{A \ar[r]^\varphi \ar[d]_{\pi} & B \\ A/\text{Ker}(\varphi) \ar@{.>}[r]^-{\theta}& \text{Im}(\varphi) \ar@{^{(}->}[u]^i }\end{xy}$  }}</span><br/>Where {{M|\theta}} is an [[group isomorphism|isomorphism]].
+|above=<div style="overflow:hidden;"><span style="font-size:1.3em;">{{M|1= $\begin{xy}\xymatrix{A \ar[r]^\varphi \ar[d]_{\pi} & B \\ A/\text{Ker}(\varphi) \ar@{.>}[r]^-{\theta}& \text{Im}(\varphi) \ar@{^{(}->}[u]^i }\end{xy}$  }}</span></div>Where {{M|\theta}} is an [[group isomorphism|isomorphism]].
 |header1=Properties
 |data1=something

Properties
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.
something