Core-free subgroup

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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:

• ${\bigcap }_{x}xH{x}^{-1}$ 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 $g(xH)\ne xH$.

Stronger properties=

• Proper malnormal subgroup
• Proper TI-subgroup

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.