This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
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This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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This term was introduced by: Wang
Wang introduced the notion of c-normal subgroup in his paper C -Normality of Groups and Its Properties.
Definition with symbols
A subgroup of a group is termed c-normal if there is a normal subgroup such that the intersection of and lies inside the normal core of , and such that .
Relation with other properties
- Normal subgroup: we can take to be the whole group .
- Retract: We can take to be a normal complement to .
Intermediate subgroup condition
YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
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If is c-normal in , with being a normal subgroup that shows it, then is also c-normal in any intermediate subgroup , and further, the normal subgroup that does the trick is . This follows because:
(the first step is the famous modular property of groups and can be proved easily.)
- C-Normality of Groups and its properties by Wang, Journal of Algebra, Volume 180, Number 3, 1 March 1996, pp. 954-965(12)