Difference between revisions of "Subgroup"
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− | A '''subgroup''' is a [[group]] contained in another. Specifically, let <math>H</math> and <math>G</math> be groups. We say that <math>H</math> is a subgroup of <math>G</math> if the [[element]]s of <math>H</math> | + | A '''subgroup''' is a [[group]] contained in another. Specifically, let <math>H</math> and <math>G</math> be groups. We say that <math>H</math> is a subgroup of <math>G</math> if the [[element]]s of <math>H</math> constitute a [[subset]] of the [[set]] of elements of <math>G</math> and the group law on <math>H</math> agrees with group law on <math>G</math> where both are defined. We may denote this by <math>H \subseteq G</math> or <math>H \le G</math>. |
We say that <math>H</math> is a ''proper subgroup'' of <math>G</math> if <math>H \neq G</math>. | We say that <math>H</math> is a ''proper subgroup'' of <math>G</math> if <math>H \neq G</math>. |
Latest revision as of 20:37, 7 May 2008
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A subgroup is a group contained in another. Specifically, let and
be groups. 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 denote this by
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. The set consisting of the identity element of a group is the smallest subgroup of that group.
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
.