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In mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup of ''G'' if the restriction of ∗ to is a group operation on ''H''. This is usually denoted , read as "''H'' is a subgroup of ''G''". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group ''G'' is a subgroup ''H'' which is a proper subset of ''G'' (i.e. ). This is usually represented notationally by , read as "''H'' is a proper subgroup of ''G''". Some authors also exclude the trivial group from being proper (i.e. ).〔Hungerford (1974), p. 32〕〔Artin (2011), p. 43〕 If ''H'' is a subgroup of ''G'', then ''G'' is sometimes called an overgroup of ''H''. The same definitions apply more generally when ''G'' is an arbitrary semigroup, but this article will only deal with subgroups of groups. The group ''G'' is sometimes denoted by the ordered pair , usually to emphasize the operation ∗ when ''G'' carries multiple algebraic or other structures. This article will write ''ab'' for , as is usual. ==Basic properties of subgroups== *A subset ''H'' of the group ''G'' is a subgroup of ''G'' if and only if it is nonempty and closed under products and inverses. (The closure conditions mean the following: whenever ''a'' and ''b'' are in ''H'', then ''ab'' and ''a''−1 are also in ''H''. These two conditions can be combined into one equivalent condition: whenever ''a'' and ''b'' are in ''H'', then ''ab''−1 is also in ''H''.) In the case that ''H'' is finite, then ''H'' is a subgroup if and only if ''H'' is closed under products. (In this case, every element ''a'' of ''H'' generates a finite cyclic subgroup of ''H'', and the inverse of ''a'' is then ''a''−1 = ''a''''n'' − 1, where ''n'' is the order of ''a''.) *The above condition can be stated in terms of a homomorphism; that is, ''H'' is a subgroup of a group ''G'' if and only if ''H'' is a subset of ''G'' and there is an inclusion homomorphism (i.e., i(''a'') = ''a'' for every ''a'') from ''H'' to ''G''. *The identity of a subgroup is the identity of the group: if ''G'' is a group with identity ''e''''G'', and ''H'' is a subgroup of ''G'' with identity ''e''''H'', then ''e''''H'' = ''e''''G''. *The inverse of an element in a subgroup is the inverse of the element in the group: if ''H'' is a subgroup of a group ''G'', and ''a'' and ''b'' are elements of ''H'' such that ''ab'' = ''ba'' = ''e''''H'', then ''ab'' = ''ba'' = ''e''''G''. *The intersection of subgroups ''A'' and ''B'' is again a subgroup.〔Jacobson (2009), p. 41〕 The union of subgroups ''A'' and ''B'' is a subgroup if and only if either ''A'' or ''B'' contains the other, since for example 2 and 3 are in the union of 2Z and 3Z but their sum 5 is not. Another example is the union of the x-axis and the y-axis in the plane (with the addition operation); each of these objects is a subgroup but their union is not. This also serves as an example of two subgroups, whose intersection is precisely the identity. *If ''S'' is a subset of ''G'', then there exists a minimum subgroup containing ''S'', which can be found by taking the intersection of all of subgroups containing ''S''; it is denoted by <''S''> and is said to be the subgroup generated by ''S''. An element of ''G'' is in <''S''> if and only if it is a finite product of elements of ''S'' and their inverses. *Every element ''a'' of a group ''G'' generates the cyclic subgroup <''a''>. If <''a''> is isomorphic to Z/''n''Z for some positive integer ''n'', then ''n'' is the smallest positive integer for which ''a''''n'' = ''e'', and ''n'' is called the ''order'' of ''a''. If <''a''> is isomorphic to Z, then ''a'' is said to have ''infinite order''. *The subgroups of any given group form a complete lattice under inclusion, called the lattice of subgroups. (While the infimum here is the usual set-theoretic intersection, the supremum of a set of subgroups is the subgroup ''generated by'' the set-theoretic union of the subgroups, not the set-theoretic union itself.) If ''e'' is the identity of ''G'', then the trivial group is the minimum subgroup of ''G'', while the maximum subgroup is the group ''G'' itself. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「subgroup」の詳細全文を読む スポンサード リンク
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