Group satisfying subnormal join property
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Definition
A group is said to satisfy the subnormal join property if it satisfies the following equivalent conditions:
- The join (i.e., subgroup generated) of two Subnormal subgroup (?)s of the group is again subnormal.
- The join of a finite collection of subnormal subgroups of the group is again subnormal.
- The commutator of any two subnormal subgroups of the group is again subnormal.
Relation with other properties
Stronger properties
- Group satisfying generalized subnormal join property
- Nilpotent group
- Group in which every subgroup is subnormal
- Simple group
- T-group
- Group with nilpotent commutator subgroup: For full proof, refer: Nilpotent commutator subgroup implies subnormal join property
- Supersolvable group
- Finite group: For full proof, refer: Finite implies subnormal join property
- Slender group: For full proof, refer: Slender implies subnormal join property
- Group satisfying ascending chain condition on subnormal subgroups: For full proof, refer: Ascending chain condition on subnormal subgroups implies subnormal join property
References
Textbook references
- A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, More info, Page 388 (definition in paragraph)