Core-free subgroup

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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.

Relation with other properties

Stronger properties

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.