Subgroup
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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 .