K implies Frattini-free
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Let be a K-group. Then the Frattini subgroup is trivial.
Definitions for K-group
A K-group is a group wih the property that every subgroup has a lattice complement. That is, a group is a K-group if for any subgroup of , there is a subgroup such that ∩ is trivial and the subgroup generated by and is the whole of .
Definitions for Frattini-free group
A Frattini-free group is a group with the property that the intersection of all its maximal subgroups (called its Frattini subgroup) is trivial. Equivalently, for any nontrivial element, there is a maximal subgroup not containing that element.
- A K-group has the property that every cyclic subgroup has a lattice complement.
- The set of subgroups that do not contain the generator of the cyclic subgroup, ordered by subgroup inclusion, satisfies the conditions needed for Zorn's lemma, and hence there is a maximal element in this
- Any subgroup maximal with respect to not containing this element, must be a maximal subgroup of the group.
- Thus, given any element, there is a maximal subgroup not containing it.
Proof with symbols
Let be a K-group and a nontrivial element of . We need to show that there is a maximal subgroup of not containing .
Let be the cyclic sugbroup generated by . Since is a K-group, there exists a proper subgroup of such that ∩ is trivial and the subgroup generated by and is the whole of .
Consider the family of subgroups of that contain do not contain . Under subgroup inclusion, this family forms a partially ordered set that satisfies the conditions for Zorn's lemma. Hence, by Zorn's lemma, there exists a subgroup maximal with respect to the property of not containing .
Since along with generates , along with also generates . Hence, any proper subgroup of containing must not contain . Since is maximal among such subgroups, is a maximal subgroup.
Thus, starting with an arbitrary nontrivial in , we have produced a maximal subgroup not containing .