,Uses,"Fact about" "B. H. Neumann's lemma",,"Left coset of a subgroup; ?,Subgroup of finite index; ?,Right coset of a subgroup; ?" "Bound on double coset index in terms of orders of group and subgroup",,"Double coset of a pair of subgroups; ?,Double coset index of a subgroup; ?,Subgroup of finite index; ?,Malnormal subgroup; ?,Normal subgroup; ?,Frobenius subgroup; ?" "Closed subgroup of finite index implies open",,"Closed subgroup of finite index; 1,Closed subgroup of semitopological group; 1,Subgroup of finite index; 2,Open subgroup; 1" "Compact implies every open subgroup has finite index","Left cosets partition a group","Compact semitopological group; 1,Open subgroup; 1,Subgroup of finite index; 2,Compact group; 1" "Connected implies no proper closed subgroup of finite index","Closed subgroup of finite index implies open","Connected topological group; 1,Closed subgroup of semitopological group; 2,Subgroup of finite index; 2" "Finite index implies completely divisibility-closed","Poincare's theorem,Subgroup of finite group implies completely divisibility-closed,Divisibility-closedness satisfies inverse image condition","Subgroup of finite index; 2,Completely divisibility-closed subgroup; 2" "Finite index implies powering-invariant","Poincare's theorem,Normal of finite index implies quotient-powering-invariant,Finite implies powering-invariant,Powering-invariant over quotient-powering-invariant implies powering-invariant","Subgroup of finite index; 2,Powering-invariant subgroup; 2" "Finite index in finite double coset index implies finite double coset index",,"Composition operator; ?,Subgroup of finite index; ?,Subgroup of finite double coset index; ?,Double coset index of a subgroup; ?" "Finite index not implies local powering-invariant",,"Subgroup of finite index; 2,Local powering-invariant subgroup; 2" "Finitely generated implies every subgroup of finite index has finitely many automorphic subgroups","Finitely generated implies finitely many homomorphisms to any finite group,Finitely many homomorphisms to any finite group implies every subgroup of finite index has finitely many automorphic subgroups","Finitely generated group; ?,Subgroup of finite index; ?,Subgroup having finitely many automorphic subgroups; ?,Subgroup of finite index in finitely generated group; 2,Subgroup having finitely many automorphic subgroups of finitely generated group; 3,Finitely generated group; 2,Group in which every subgroup of finite index has finitely many automorphic subgroups; 2" "Index is multiplicative","Subgroup containment implies coset containment","Subgroup of finite index; 1,Transitive subgroup property; 2,Index of a subgroup; 1" "Poincare's theorem","Group acts on left coset space of subgroup by left multiplication,First isomorphism theorem,Lagrange's theorem,Index is multiplicative","Subgroup of finite index; 1,Normal core-closed subgroup property; 2" "Schreier's lemma",,"Left transversal of a subgroup; ?,Generating set of a group; ?,Subgroup of finite index; 2,Finitely generated group; 2,Subgroup of finite index in finitely generated group; 1" "Subgroup of finite index has a left transversal that is also a right transversal","Subgroup of finite group has a left transversal that is also a right transversal,Poincare's theorem","Subgroup of finite index; 2,Subgroup having a left transversal that is also a right transversal; 2,Left transversal of a subgroup; ?,Right transversal of a subgroup; ?" "Subgroup of finite index need not be closed in algebraic group",,"Subgroup of finite index; 2"