Central factor satisfies image condition
This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor) satisfying a subgroup metaproperty (i.e., image condition)
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Statement with symbols
Further information: Central factor
A subgroup of a group is termed a central factor of , where denotes the centralizer of in .
- Every inner automorphism of restricts to an inner automorphism of .
- Transitive normality satisfies image condition
- SCAB satisfies image condition
- Direct factor satisfies image condition
Proof in terms of centralizers
Given: A central factor of a group . A surjective homomorphism .
To prove: .
Proof: By the definition of homomorphism, if two elements commute in , their images commute in . Thus, the definition of centralizer yields:
Taking the product of both sides with yields:
By the definition of homomorphism, the left side is the same as , which is (since is a central factor of ). by the assumption of surjectivity, so we get:
completing the proof.