Transitive and transfer condition implies finite-relative-intersection-closed
Suppose is a Transitive subgroup property (?) satisfying the Transfer condition (?). Then, is a Finite-relative-intersection-closed subgroup property (?).
Transitive subgroup property
Further information: Transitive subgroup property
A subgroup property is termed transitive if whenever are groups such that has property in , and has property in , we have that has property in .
Further information: Transfer condition
A subgroup property is said to satisfy the transfer condition if whenever are subgroups such that satisfies property in , we have that satisfies property in .
Finite-relative-intersection-closed subgroup property
Further information: Finite-relative-intersection-closed subgroup property
A subgroup property is termed finite-relative-intersection-closed if whenever are subgroups such that satisfies property in and satisfies property in some subgroup containing both and , then satisfies property in .
- Subnormal subgroup: Subnormality is finite-relative-intersection-closed follows from the facts that subnormality is transitive and subnormality satisfies transfer condition.
Given: A group with subgroups such that satisfies property in and satisfies property in some subgroup containing both and . is both transitive and satisfies the transfer condition.
To prove: satisfies property in .
- (Given data used: satisfies in , satisfies transfer condition): satisfies property in : satisfies property in , and is a subgroup of . Since satisfies the transfer condition, we have that satisfies property in .
- (Given data used: is transitive, satisfies in ): By the previous step, we have satisfies property in , and satisfies property in . Since is transitive, we obtain that satisfies in .