# Core-free subgroup

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]

This is an opposite of normality

## History

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## Definition

### Symbol-free definition

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

• Its normal core (viz the intersection of al its conjugates) is trivial
• The action of the group on the coset space is effective (that is, every element of the group acts nontrivially on the coset space)

### Definition with symbols

A subgroup $H$ of a group $G$ is termed core-free if it satisfies the following equivalent conditions:

• $\bigcap_x xHx^{-1}$ is the trivial subgroup.
• The action of $G$ by left multiplication on the coset space $G/H$ is effective, that is, for every $g \in G$, there is a coset $xH$ such that $g(xH) \ne xH$.

## Formalisms

### First-order description

This subgroup property is a first-order subgroup property, viz., it has a first-order description in the theory of groups.
View a complete list of first-order subgroup properties

A subgroup $H$ is core-free in a group $G$ if: $\forall x \in H . (\forall g \in G, gxg^{-1} \in H), x = e$

Essentially, we are using the fact that the normal core can be described using a first-order condition.

## Facts

Given a subgroup $H$ of $G$, $G$ acts transitively on the coset space of $H$.

Conversely any transitive group action can be modelled as an action on the coset space of a subgroup.

Thus, we have the following: $G$ has a core-free subgroup of index $n$ if and only if it can be embedded as a transitive subgroup of the symmetric group on $n$ elements.

## Metaproperties

### Lower hereditariness

Core-freeness is a lower hereditary property, that is, any subgroup of a core-free subgroup is again core-free. Moreover, the trivial subgroup is always core-free.

From lower hereditariness, it follows that core-freeness is transitive and that an intersection of core-free subgroups is core-free.

### Upper hereditariness

Core-freeness is also upper hereditary: any core-free subgroup of a subgroup is core-free in the whole group.

## Relation with group actions

### Primitive groups

If a group has a maximal subgroup that is also core-free, then it is termed a primitive group. This is equivalent to the other definition: the group is primitive if it has an effective primitive group action.