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

The historical roots of this term, viz how the term and the concept were developed, are missing from this article. If you have any idea or knowledge, please contribute right now by editing this section. To learn more about what goes into the History section, click here

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.