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In mathematics, one method of defining a group is by a presentation. One specifies a set ''S'' of generators so that every element of the group can be written as a product of powers of some of these generators, and a set ''R'' of relations among those generators. We then say ''G'' has presentation : Informally, ''G'' has the above presentation if it is the "freest group" generated by ''S'' subject only to the relations ''R''. Formally, the group ''G'' is said to have the above presentation if it is isomorphic to the quotient of a free group on ''S'' by the normal subgroup generated by the relations ''R''. As a simple example, the cyclic group of order ''n'' has the presentation : where 1 is the group identity. This may be written equivalently as : since terms that don't include an equals sign are taken to be equal to the group identity. Such terms are called relators, distinguishing them from the relations that include an equals sign. Every group has a presentation, and in fact many different presentations; a presentation is often the most compact way of describing the structure of the group. A closely related but different concept is that of an absolute presentation of a group. == Background == A free group on a set ''S'' is a group where each element can be ''uniquely'' described as a finite length product of the form: : where the ''si'' are elements of S, adjacent ''si'' are distinct, and ''ai'' are non-zero integers (but ''n'' may be zero). In less formal terms, the group consists of words in the generators ''and their inverses'', subject only to canceling a generator with its inverse. If ''G'' is any group, and ''S'' is a generating subset of ''G'', then every element of ''G'' is also of the above form; but in general, these products will not uniquely describe an element of ''G''. For example, the dihedral group D8 of order sixteen can be generated by a rotation, ''r'', of order 8; and a flip, ''f'', of order 2; and certainly any element of D8 is a product of ''r''s and ''f''s. However, we have, for example, , , etc., so such products are not unique in D8. Each such product equivalence can be expressed as an equality to the identity, such as :''rfrf'' = 1 :''r''8 = 1 :''f''2 = 1. Informally, we can consider these products on the left hand side as being elements of the free group , and can consider the subgroup ''R'' of ''F'' which is generated by these strings; each of which would also be equivalent to 1 when considered as products in D8. If we then let ''N'' be the subgroup of ''F'' generated by all conjugates ''x''−1''Rx'' of ''R'', then it is straightforward to show that every element of ''N'' is a finite product ''x''1−1''r''1''x''1 ... ''xm''−1''rm'' ''xm'' of members of such conjugates. It follows that ''N'' is a normal subgroup of ''F''; and that each element of ''N'', when considered as a product in D8, will also evaluate to 1. Thus D8 is isomorphic to the quotient group . We then say that D8 has presentation : 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「presentation of a group」の詳細全文を読む スポンサード リンク
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