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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 .
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 .