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.