, in the realm of group theory
, a group
is termed a complemented group
or a K group
if every subgroup
of it has a lattice theoretic complement
. That is, G
is a complemented group if for every subgroup H
there is a subgroup L
that intersects H
trivially and that, along with H
. Equivalently, G
is complemented if and only if the lattice of subgroups
is a complemented lattice
Every finite simple group is a complemented group. The proof of this requires the classification of finite simple groups.
An example of a group that is not complemented 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.