Transitive normality satisfies image condition
This article gives the statement, and possibly proof, of a subgroup property (i.e., transitively normal subgroup) satisfying a subgroup metaproperty (i.e., image condition)
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Statement with symbols
Suppose is a transitively normal subgroup of a group . Suppose is a surjective homomorphism of groups. Then, is a transitively normal subgroup of .
Similar facts about similar properties
- Central factor satisfies image condition
- SCAB satisfies image condition
- Direct factor satisfies image condition
- Centrality satisfies image condition
Related facts about transitively normal subgroups
Given: A group , a subgroup . A surjective homomorphism of groups. . is a normal subgroup of .
To prove: is normal in .
- is normal in and : Let be the restriction of . Then, is a surjective homomorphism by definition, and fact (1) yields that is normal in . Further, clearly surjects to , since is surjective. But by definition, so is normal in and .
- is normal in : From the previous step, is normal in . By assumption, is transitively normal in . Thus, must be normal in .
- is normal in </math>K</math>: This follows from fact (2).