# Max-core

From Groupprops

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]

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## 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 of a group is termed a **max-core** if it satisfies the following equivalent conditions:

- There is a maximal subgroup of such that is the normal core of
- There is a primitive group action of <amth>G</math> such that the kernel of is

## 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.