Max-core

Symbol-free definition
A subgroup of a group is termed a max-core if it satisfies the following equivalent conditions:


 * It occurs as the normal core of a maximal subgroup
 * it occurs as the kernel of a primitive group action

Definition with symbols
A subgroup $$N$$ of a group $$G$$ is termed a max-core if it satisfies the following equivalent conditions:


 * There is a maximal subgroup $$M$$ of $$G$$ such that $$N$$ is the normal core of $$M$$
 * There is a primitive group action $$\alpha$$ of G such that the kernel of $$\alpha$$ is $$N$$

Stronger properties

 * Maximal normal subgroup when the group involved has the property that every subgroup is contained in a maximal subgroup

Weaker properties

 * Subgroup containing the Frattini subgroup

Related group properties
A group is primitive if and only if the trivial subgroup is a max-core.