Complemented group

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":
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" = { "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 . 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".