|
In mathematics, in the realm of group theory, the term complemented group is used in two distinct, but similar ways. In , a complemented group is one in which every subgroup has a group-theoretic complement. Such groups are called completely factorizable groups in the Russian literature, following and . The following are equivalent for any finite group ''G'': * ''G'' is complemented * ''G'' is a subgroup of a direct product of groups of square-free order (group theory) (a special type of Z-group) * ''G'' is a supersolvable group with elementary abelian Sylow subgroups (a special type of A-group), . Later, in , a group is said to be complemented if the lattice of subgroups is a complemented lattice, that is, if for every subgroup ''H'' there is a subgroup ''K'' such that ''H''∩''K''=1 and ⟨''H'',''K''⟩ is the whole group. Hall's definition required in addition that ''H'' and ''K'' permute, that is, that ''HK'' = form a subgroup. Such groups are also called K-groups in the Italian and lattice theoretic literature, such as . The Frattini subgroup of a K-group is trivial; if a group has a core-free maximal subgroup that is a K-group, then it itself is a K-group; hence subgroups of K-groups need not be K-groups, but quotient groups and direct products of K-groups are K-groups, . In it is shown that every finite simple group is a complemented group. Note that in the classification of finite simple groups, ''K''-group is more used to mean a group whose proper subgroups only have composition factors amongst the known finite simple groups. An example of a group that is not complemented (in either sense) is the cyclic group of order ''p''2, where ''p'' is a prime number. This group only has one nontrivial subgroup ''H'', the cyclic group of order ''p'', so there can be no other subgroup ''L'' to be the complement of ''H''. ==References== * * * * * * 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「Complemented group」の詳細全文を読む スポンサード リンク
|