603: Injective Group Homomorphism Is 'Groups - Homomorphisms' Isomorphism onto Range

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description/proof of that injective group homomorphism is 'groups - homomorphisms' isomorphism onto range

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Topics

About: group

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Proof

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Starting Context

• The reader knows a definition of group.
• The reader knows a definition of Injection.
• The reader knows a definition of %structure kind name% homomorphism.
• The reader knows a definition of %category name% isomorphism.
• The reader admits the proposition that for any group homomorphism, the range of the homomorphism is a subgroup of the codomain.
• The reader admits the proposition that any bijective group homomorphism is a 'groups - homomorphisms' isomorphism.

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Target Context

• The reader will have a description and a proof of the proposition that any injective group homomorphism is a 'groups - homomorphisms' isomorphism onto the range of the homomorphism.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$G_1$: $\in \{\text{ the groups }\}$
$G_2$: $\in \{\text{ the groups }\}$
$f$: $G_1\to G_2$, $\in \{\text{ the injections }\}\cap \{\text{ the group homomorphisms }\}$
$f^{\prime }$: $G_1\to f(G_1)$, $g\mapsto f(g)$
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Statements:
$f(G_1)\in \{\text{ the subgroups of }{G}_2\}$
$\wedge$
$f^{\prime }\in \{\text{ the 'groups - homomorphisms' isomorphisms }\}$
//

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2: Natural Language Description

For any groups, $G_1,G_2$, and any injective group homomorphism, $f:G_1\to G_2$, $f(G_1)\subseteq G_2$ is a subgroup of $G_2$ and $f^{\prime }:G_1\to f(G_1)$ is a 'groups - homomorphisms' isomorphism.

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3: Proof

$f(G_1)$ is a subgroup of $G_2$, by the proposition that for any group homomorphism, the range of the homomorphism is a subgroup of the codomain.

$f^{\prime }$ is a bijective group homomorphism.

So, $f^{\prime }$ is a 'groups - homomorphisms' isomorphism, by the proposition that any bijective group homomorphism is a 'groups - homomorphisms' isomorphism.

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References

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