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A non-empty subset of a group which itself is a group with respect to the operation defined on . A subset of a group is a subgroup of if and only if: 1) contains the product of any two elements from ; and 2) contains together with any element the inverse . In the cases of finite and periodic groups, condition 2) is superfluous.

The subset of a group G consisting of the element 1 only is clearly a subgroup; it is called the unit subgroup of and is usually denoted by . Also, itself is a subgroup. A subgroup different from is called a proper subgroup of . A proper subgroup of an infinite group can be isomorphic to the group itself. The group itself and the subgroup are called improper subgroups of , while all the others are called proper ones.

The set-theoretic intersection of any two (or any set of) subgroups of a group is a subgroup of . The intersection of all subgroups of containing all elements of a certain non-empty set is called the subgroup generated by the set and is denoted by . If consists of one element , then is called the cyclic subgroup of the element . A group that coincides with one of its cyclic subgroups is called a cyclic group.

A set-theoretic union of subgroups is, in general, not a subgroup. By the join of subgroups , , one means the subgroup generated by the union of the sets .

The product of two subsets and of a group is the set consisting of all possible (different) products , where , . The product of two subgroups is a subgroup if and only if , and in that case the product coincides with the subgroup generated by and (i.e. with the join of and ).

A homomorphic image of a subgroup is a subgroup. If a group is isomorphic to a subgroup of a group , one says that can be imbedded in (as groups). If one is given two groups and each of them is isomorphic to a proper subgroup of the other, it does not necessarily follow that these groups themselves are isomorphic (cf. also Homomorphism; Isomorphism).


Comments

References

[a1] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[a2] M. Hall jr., "The theory of groups" , Macmillan (1959) pp. 124
How to Cite This Entry:
Subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subgroup&oldid=33209
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article