Complemented group

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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:

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.

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