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- Complemented_group abstract "In mathematics, in the realm of group theory, the term complemented group is used in two distinct, but similar ways.In (Hall 1937), 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 (Baeva 1953) and (Černikov 1953). 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), (Hall 1937, Theorem 1 and 2).Later, in (Zacher 1953), 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 = { hk : h in H, k in K } form a subgroup. Such groups are also called K-groups in the Italian and lattice theoretic literature, such as (Schmidt 1994, pp. 114–121, Chapter 3.1). 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, (Schmidt 1994, pp. 115–116). In (Costantini & Zacher 2004) 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 p2, 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.".
- Complemented_group wikiPageExternalLink item?id=RSMUP_1953__22__113_0.
- Complemented_group wikiPageID "5780991".
- Complemented_group wikiPageLength "4051".
- Complemented_group wikiPageOutDegree "27".
- Complemented_group wikiPageRevisionID "656816245".
- Complemented_group wikiPageWikiLink A-group.
- Complemented_group wikiPageWikiLink Category:Properties_of_groups.
- Complemented_group wikiPageWikiLink Classification_of_finite_simple_groups.
- Complemented_group wikiPageWikiLink Complement_(group_theory).
- Complemented_group wikiPageWikiLink Complemented_lattice.
- Complemented_group wikiPageWikiLink Core_(group_theory).
- Complemented_group wikiPageWikiLink Cyclic_group.
- Complemented_group wikiPageWikiLink Direct_product_of_groups.
- Complemented_group wikiPageWikiLink Elementary_abelian_group.
- Complemented_group wikiPageWikiLink Finite_group.
- Complemented_group wikiPageWikiLink Frattini_subgroup.
- Complemented_group wikiPageWikiLink Group_(mathematics).
- Complemented_group wikiPageWikiLink Group_theory.
- Complemented_group wikiPageWikiLink Lattice_of_subgroups.
- Complemented_group wikiPageWikiLink List_of_finite_simple_groups.
- Complemented_group wikiPageWikiLink London_Mathematical_Society.
- Complemented_group wikiPageWikiLink Mathematics.
- Complemented_group wikiPageWikiLink Maximal_subgroup.
- Complemented_group wikiPageWikiLink Order_(group_theory).
- Complemented_group wikiPageWikiLink Pacific_Journal_of_Mathematics.
- Complemented_group wikiPageWikiLink Prime_number.
- Complemented_group wikiPageWikiLink Quotient_group.
- Complemented_group wikiPageWikiLink Rendiconti_del_Seminario_Matematico_della_Università_di_Padova.
- Complemented_group wikiPageWikiLink Subgroup.
- Complemented_group wikiPageWikiLink Supersolvable_group.
- Complemented_group wikiPageWikiLink Sylow_theorems.
- Complemented_group wikiPageWikiLink Z-group.
- Complemented_group wikiPageWikiLinkText "complemented group".
- Complemented_group wikiPageUsesTemplate Template:Abstract-algebra-stub.
- Complemented_group wikiPageUsesTemplate Template:Citation.
- Complemented_group wikiPageUsesTemplate Template:Harv.
- Complemented_group subject Category:Properties_of_groups.
- Complemented_group type Property.
- Complemented_group comment "In mathematics, in the realm of group theory, the term complemented group is used in two distinct, but similar ways.In (Hall 1937), 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 (Baeva 1953) and (Černikov 1953).".
- Complemented_group label "Complemented group".
- Complemented_group sameAs Q5156432.
- Complemented_group sameAs m.0f46mg.
- Complemented_group sameAs Q5156432.
- Complemented_group wasDerivedFrom Complemented_group?oldid=656816245.
- Complemented_group isPrimaryTopicOf Complemented_group.