# Semantic search

B. H. Neumann's lemma, Bound on double coset index in terms of orders of group and subgroup, Closed subgroup of finite index implies open, Compact implies every open subgroup has finite index, Connected implies no proper closed subgroup of finite index, Finite index implies completely divisibility-closed, Finite index implies powering-invariant, Finite index in finite double coset index implies finite double coset index, Finite index not implies local powering-invariant, Finitely generated implies every subgroup of finite index has finitely many automorphic subgroups, Index is multiplicative, Poincare's theorem, Schreier's lemma, Subgroup of finite index has a left transversal that is also a right transversal, Subgroup of finite index need not be closed in algebraic group