Tour:Subgroup generated by a subset
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This article adapts material from the main article: subgroup generated by a subset
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PREREQUISITES: Intersection of subgroups, and generating set of a group.
WHAT YOU NEED TO DO: Understand the equivalent formulations of subgroup generated by a subset.
Contents
Definition
Symbol-free definition
Given a group, and a subset of the group, the subgroup generated by that subset is defined in the following equivalent ways:
- It is the intersection of all subgroups containing that subset
- It is a subgroup containing that subset, such that the subset is a generating set for the subgroup
Definition with symbols
Given a group and a subset
of
, the subgroup generated by
is defined in the following equivalent ways:
- It is the intersection of all subgroups of
containing
- It is a subgroup
such that
and
is a generating set for
(Recall the fact that an intersection of subgroups is always a subgroup).
The subgroup generated by a subset is denoted
.
Examples
Extreme examples
- If a subset
of a group
is a subgroup, then
equals the subgroup generated by
.
- The subgroup generated by the empty subset is the trivial subgroup: it comprises only the identity element.
Other generic examples
- The subgroup generated by a single element is the set of all its powers. In other words, the subgroup generated by an element
is the set of all elements expressible as
. This is also termed the cyclic subgroup generated by
.
Examples in Abelian groups
- In the group of integers under addition, the subgroup generated by the integers
and
is the subgroup of even integers.
- In the group of rational numbers under addition, the subgroup generated by the rational number
is the group of integers and half-integers under addition.
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