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
- 1 History
- 2 Definition
- 3 Formalisms
- 4 Relation with other properties
- 5 Facts
- 6 Metaproperties
- 7 Relation with group actions
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- 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 of a group is termed core-free if it satisfies the following equivalent conditions:
- is the trivial subgroup.
- The action of by left multiplication on the coset space is effective, that is, for every , there is a coset such that .
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 is core-free in a group if:
Essentially, we are using the fact that the normal core can be described using a first-order condition.
Relation with other properties
Given a subgroup of , acts transitively on the coset space of .
Conversely any transitive group action can be modelled as an action on the coset space of a subgroup.
Thus, we have the following:
has a core-free subgroup of index if and only if it can be embedded as a transitive subgroup of the symmetric group on elements.
From lower hereditariness, it follows that core-freeness is transitive and that an intersection of core-free subgroups is core-free.
Core-freeness is also upper hereditary: any core-free subgroup of a subgroup is core-free in the whole group.
Relation with group actions
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.