Max-core

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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Definition

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 <amth>G</math> such that the kernel of $\alpha$ is $N$

Relation with other properties

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.