|
In mathematics, the term socle has several related meanings. ==Socle of a group== In the context of group theory, the socle of a group ''G'', denoted soc(''G''), is the subgroup generated by the minimal normal subgroups of ''G''. It can happen that a group has no minimal non-trivial normal subgroup (that is, every non-trivial normal subgroup properly contains another such subgroup) and in that case the socle is defined to be the subgroup generated by the identity. The socle is a direct product of minimal normal subgroups. As an example, consider the cyclic group Z12 with generator ''u'', which has two minimal normal subgroups, one generated by ''u'' 4 (which gives a normal subgroup with 3 elements) and the other by ''u'' 6 (which gives a normal subgroup with 2 elements). Thus the socle of Z12 is the group generated by ''u'' 4 and ''u'' 6, which is just the group generated by ''u'' 2. The socle is a characteristic subgroup, and hence a normal subgroup. It is not necessarily transitively normal, however. If a group G is a finite solvable group, then the socle can be expressed as a product of elementary abelian p-groups. Thus, in this case, it is just a product of copies of Z/pZ for various p where the same p may occur multiple times in the product. 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「Socle (mathematics)」の詳細全文を読む スポンサード リンク
|