Difference between revisions of "Subgroup"
(New page: {{stub}} A '''subgroup''' is a group contained in another. Specifically, let <math>H</math> and <math>G</math> be groups (with group laws written multiplicatively). We say that <mat...) |
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In a group <math>G</math>, the intersection of a family of subgroups of <math>G</math> is a subgroup of <math>G</math>. Thus for any collection <math>X</math> of elements of <math>G</math>, there exists a smallest subgroup containing these elements. This is called the subgroup generated by <math>X</math>. | In a group <math>G</math>, the intersection of a family of subgroups of <math>G</math> is a subgroup of <math>G</math>. Thus for any collection <math>X</math> of elements of <math>G</math>, there exists a smallest subgroup containing these elements. This is called the subgroup generated by <math>X</math>. | ||
− | In the additive group <math>\mathbb{Z}</math>, all subgroups are of the form <math>n \mathbb{Z}</math>, for some integer <math> | + | In the additive group <math>\mathbb{Z}</math>, all subgroups are of the form <math>n \mathbb{Z}</math>, for some integer <math>n</math>. In particular, for <math>n=1</math>, we have the integers themselves, and for <math>n=0</math>, we have <math>\{0\}</math>. |
== See Also == | == See Also == |
Revision as of 14:17, 19 February 2008
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A subgroup is a group contained in another. Specifically, let and
be groups (with group laws written multiplicatively). We say that
is a subgroup of
if the elements of
constitute a subset of the set of elements of
, and the group law on
agrees with group law on
where both are defined. We may also write
or
.
We say that is a proper subgroup of
if
.
Examples
In the additive group , shown below,
there are three subgroups : the group itself,
, and the group
, shown below. This last subgroup is isomorphic to the additive group
.
Every group is the largest subgroup of itself. In a group with identity ,
is the smallest subgroup.
In a group , the intersection of a family of subgroups of
is a subgroup of
. Thus for any collection
of elements of
, there exists a smallest subgroup containing these elements. This is called the subgroup generated by
.
In the additive group , all subgroups are of the form
, for some integer
. In particular, for
, we have the integers themselves, and for
, we have
.