B. H. Neumann's lemma

This article describes an easy-to-prove fact about basic notions in group theory, that is not very well-known or important in itself
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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, $gH=(gHg^{-1})g$), and since conjugate subgroups have the same index in the whole group, we can replace left coset by right coset above.

In symbols

If a group $G$ can be written as a union of left cosets:

$G=g_{1}H_{1}\cup g_{2}H_{2}\cup \dots \cup g_{n}H_{n}$

then $G$ is the union of those $g_{i}H_{i}$ for which $H_{i}$ is a subgroup of finite index in $G$. In other words, the cosets of subgroups of infinite index are redundant.

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 three proper subgroups is the whole group 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

The geometry of cosets

• Coset containment implies subgroup containment
• Nonempty intersection of cosets is coset of intersection