Monomorphism iff injective in the category of groups
This article gives a proof/explanation of the equivalence of multiple definitions for the term injective homomorphism
View a complete list of pages giving proofs of equivalence of definitions
The following are equivalent for a homomorphism of groups :
- is injective as a set map.
- is a monomorphism with respect to the category of groups: For any homomorphisms from any group , .
Injective homomorphism implies monomorphism
This follows simply by thinking of the maps as set maps. In general, for any concrete category, any injective homomorphism is a monomorphism.
Monomorphism implies injective homomorphism
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Given: is a monomorphism: For any homomorphisms from any group , .
To prove: is injective, i.e., the kernel of is the trivial subgroup of .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||Let be the kernel of . Let be natural inclusion of in and be the trivial homomorphism from to .|
|2||as homomorphisms from to , since both and are trivial homomorphisms from to .||Step (1)||By Step (1), is the kernel of and is its natural inclusion in , so the composite is trivial. is trivial because is trivial.|
|3||.||is a monomorphism||Step (2)||Step-given combination direct.|
|4||is trivial, i.e., the kernel of is trivial.||Steps (1), (3)||Step (3) tells us that the inclusion of in is the same as the trivial homomorphism, forcing to be trivial. This completes the proof.|