Core-free subgroup

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
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.

Stronger properties

 * Proper malnormal subgroup
 * Proper TI-subgroup

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.

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.

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.