In mathematics, 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 of G there is a subgroup L that intersects H trivially and that, along with H generates G. Equivalently, G is complemented if and only if the lattice of subgroups of G 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.

External links

• The proof that finite simple groups have complemented subgroup lattices