Directed union of subgroups is subgroup
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This article gives the statement, and possibly proof, of a basic fact in group theory.
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Statement
Verbal statement
The union of a nonempty directed set of subgroups of a group is again a subgroup.
Statement with symbols
Suppose is a group, a nonempty directed set, and is a collection of subgroups of indexed by , such that . Then, the subset of given by:
is also a subgroup of .
Definitions used
Directed set
A partially ordered set is termed directed if for any , there exists , such that .
Proof
Given: is a group, a nonempty directed set, and is a collection of subgroups of indexed by , such that .
To prove: The subset of given by:
is also a subgroup of .
Proof: We check the three conditions for a subgroup:
- Identity element: Indeed, the identity element of is in all the s, so it is in their union.
- Inverse elements: Suppose is in the union. Then, for some . Thus, (because is a subgroup). So, is in the union.
- Products: Suppose are in the union. Then, , for some . By the directedness property, there exists , such that . Thus, and . In particular, both and are in the subgroup . So, their product is in , so is in the union.