Transitive and transfer condition implies finite-intersection-closed

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Statement

A Transitive subgroup property (?) that satisfies the Transfer condition (?) is finite-intersection-closed.

Definitions used

Transitive subgroup property

Further information: Transitive subgroup property

A subgroup property p is termed transitive if whenever H \le K \le G are groups such that H satisfies property p in K and K satisfies property p in G, H also satisfies property p in G.

Transfer condition

Further information: Transfer condition

A subgroup property p is said to satisfy the transfer condition if whenever H, K \le G such that H satisfies property p in G, H \cap K satisfies property p in K.

Finite-intersection-closed subgroup property

Further information: Finite-intersection-closed subgroup property

A subgroup property p is termed finite-intersection-closed if whenever H, K are subgroups satisfying property p in G, then H \cap K also satisfies property p in G.

Facts used

  1. Transitive and transfer condition implies finite-relative-intersection-closed
  2. Finite-relative-intersection-closed implies finite-intersection-closed

Proof

Proof using given facts

The proof follows directly by piecing together facts (1) and (2).

Hands-on proof

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