Endomorphism kernel is quotient-transitive
This article gives the statement, and possibly proof, of a subgroup property (i.e., endomorphism kernel) satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property)
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Get more facts about endomorphism kernel |Get facts that use property satisfaction of endomorphism kernel | Get facts that use property satisfaction of endomorphism kernel|Get more facts about quotient-transitive subgroup property
Suppose are groups such that is an endomorphism kernel in and the quotient group is an endomorphism kernel in . (Note that it makes sense to take quotients because endomorphism kernel implies normal). Then, is an endomorphism kernel in .
Similar facts about similar properties
- Normality is quotient-transitive
- Characteristicity is quotient-transitive
- Complemented normal is quotient-transitive
Opposite facts about endomorphism kernel
- Third isomorphism theorem: This basically tells us that .
Given: Groups such that is an endomorphism kernel in and is an endomorphism kernel in .
To prove: is an endomorphism kernel in .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||Let be a subgroup of isomorphic to and let be an isomorphism between them.||is an endomorphism kernel in||Definition-direct|
|2||Let be a subgroup of isomorphic to .||is an endomorphism kernel in .||Definition-direct|
|3||is isomorphic to .||Fact (1)||, normal in , normal in (normality follows from being endomorphism kernels)||Fact-direct|
|4||is a subgroup of isomorphic to .||Steps (2), (3)||step-combination direct|
|5||is a subgroup of (and hence of ) isomorphic to||Steps (1), (2)||is an isomorphism, hence it maps subgroups to subgroups isomorphic to them. Apply this to .|
|6||is a subgroup of isomorphic to .||Steps (4), (5)||step-combination direct.|
|7||is an endomorphism kernel in .||Step (6)||definition-direct.|