In mathematics, in the realm of group theory, the term '''complemented group''' is used in two distinct, but similar ways.
In {{harv|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 {{harv|Baeva|1953}} and {{harv|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 (a special type of Z-group) * ''G'' is a supersolvable group with elementary abelian Sylow subgroups (a special type of A-group), {{harv|Hall|1937|loc=Theorem 1 and 2}}.
Later, in {{harv|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''{{hairsp}}⟩ 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 {{harv|Schmidt|1994|loc=Chapter 3.1|pp=114–121}}. 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, {{harv|Schmidt|1994|pp=115–116}}. In {{harv|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 ''p''<sup>2</sup>, 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== * {{Citation | last1=Baeva | first1=N. V. | title=Completely factorizable groups | mr=0059275 | year=1953 | journal=Doklady Akademii Nauk SSSR |series=New Series | volume=92 | pages=877–880}} <!-- Gathered from intros of several papers: Baeva is later known as Černikova, N. V., perhaps married to or mother of Černikov, N. S., who in turn was perhaps son of Černikov, S. N. ; at any rate, all three wrote on complemented groups --> * {{Citation | last1=Černikov | first1=S. N. | title=Groups with systems of complementary subgroups | mr=0059276 | year=1953 | journal=Doklady Akademii Nauk SSSR |series=New Series | volume=92 | pages=891–894}} * {{Citation | last1=Costantini | first1=Mauro | last2=Zacher | first2=Giovanni | title=The finite simple groups have complemented subgroup lattices | doi=10.2140/pjm.2004.213.245 | mr=2036918 | year=2004 | journal=Pacific Journal of Mathematics | issn=0030-8730 | volume=213 | issue=2 | pages=245–251| doi-access=free | hdl=11577/1341437 | hdl-access=free }} * {{citation | last=Hall | first=Philip | title=Complemented groups | journal=J. London Math. Soc. | volume=12 | pages=201–204 | year=1937 | issue=3 | zbl=0016.39301 | doi=10.1112/jlms/s1-12.2.201 }} *{{Citation | last1=Schmidt | first1=Roland | title=Subgroup Lattices of Groups | publisher=Walter de Gruyter | series=Expositions in Math | isbn=978-3-11-011213-9 | mr=1292462 | year=1994 | volume=14}} * {{Citation | last1=Zacher | first1=Giovanni | title=Caratterizzazione dei gruppi risolubili d'ordine finito complementati | url=http://www.numdam.org/item?id=RSMUP_1953__22__113_0 | mr=0057878 | year=1953 | journal=Rendiconti del Seminario Matematico della Università di Padova | issn=0041-8994 | volume=22 | pages=113–122}}
Category:Properties of groups
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