# B. H. Neumann's lemma

From Groupprops

## Contents

## Statement

If a group arises as a union of finitely many Left coset (?)s of (possibly same, possibly different) subgroups, then at least one of those is a coset of a Subgroup of finite index (?).

Note that since a left coset of a subgroup is a Right coset (?) of one of its conjugate subgroups (specifically, ), and since conjugate subgroups have the same index in the whole group, we can replace left coset by right coset above.

## Related facts

### Unions of subgroups

- Union of two subgroups is not a subgroup unless they are comparable
- Union of three subgroups is a subgroup implies they have index two and form a flower arrangement
- Union of n subgroups is the whole group iff the group admits one of finitely many groups as quotient
- There is no group that is a union of seven proper subgroups but not a union of fewer proper subgroups

### Related facts in group theory

- Directed union of subgroups is subgroup
- Union of all conjugates is proper
- Every group is a union of cyclic subgroups
- Every group is a union of maximal among Abelian subgroups
- Cyclic iff not a union of proper subgroups