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In mathematics, especially in the area of algebra known as group theory, the holomorph of a group is a group which simultaneously contains (copies of) the group and its automorphism group. The holomorph provides interesting examples of groups, and allows one to treat group elements and group automorphisms in a uniform context. In group theory, for a group , the holomorph of denoted can be described as a semidirect product or as a permutation group. ==Hol(''G'') as a semi-direct product== If is the automorphism group of then : where the multiplication is given by : (1 ) Typically, a semidirect product is given in the form where and are groups and is a homomorphism and where the multiplication of elements in the semi-direct product is given as : which is well defined, since and therefore . For the holomorph, and is the identity map, as such we suppress writing explicitly in the multiplication given in (1 ) above. For example, * the cyclic group of order 3 * where * with the multiplication given by: : where the exponents of are taken mod 3 and those of mod 2. Observe, for example : and note also that this group is not abelian, as , so that is a non-abelian group of order 6 which, by basic group theory, must be isomorphic to the symmetric group . 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「Holomorph (mathematics)」の詳細全文を読む スポンサード リンク
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