# Complemented group

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 (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 *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

- Baeva, N. V. (1953), "Completely factorizable groups",
*Doklady Akademii Nauk SSSR (N.S.)*,**92**: 877–880, MR 0059275 - Černikov, S. N. (1953), "Groups with systems of complementary subgroups",
*Doklady Akademii Nauk SSSR (N.S.)*,**92**: 891–894, MR 0059276 - Costantini, Mauro; Zacher, Giovanni (2004), "The finite simple groups have complemented subgroup lattices",
*Pacific Journal of Mathematics*,**213**(2): 245–251, doi:10.2140/pjm.2004.213.245, ISSN 0030-8730, MR 2036918 - Hall, Philip (1937), "Complemented groups",
*J. London Math. Soc.*,**12**: 201–204, doi:10.1112/jlms/s1-12.2.201, Zbl 0016.39301 - Schmidt, Roland (1994),
*Subgroup Lattices of Groups*, Expositions in Math,**14**, Walter de Gruyter, ISBN 978-3-11-011213-9, MR 1292462 - Zacher, Giovanni (1953), "Caratterizzazione dei gruppi risolubili d'ordine finito complementati",
*Rendiconti del Seminario Matematico della Università di Padova*,**22**: 113–122, ISSN 0041-8994, MR 0057878