C-normal subgroup

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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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History

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

Symbol-free definition

A subgroup of a group is termed c-normal if there is a normal subgroup whose product with it is the whole group and whose intersection with it lies inside its normal core.

Definition with symbols

A subgroup H of a group G is termed c-normal if there is a normal subgroup T such that the intersection of H and T lies inside the normal core of H, and such that HT = G.

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

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 H is c-normal in G, with T being a normal subgroup that shows it, then H is also c-normal in any intermediate subgroup K, and further, the normal subgroup that does the trick is T \cap K. This follows because:

H(T \cap K) = HT \cap K = G \cap K = K

(the first step is the famous modular property of groups and can be proved easily.)

References

  • C-Normality of Groups and its properties by Wang, Journal of Algebra, Volume 180, Number 3, 1 March 1996, pp. 954-965(12)