core of a subgroup
Let be a subgroup of a group .
The core (or normal interior, or normal core) of in is the intersection of all conjugates of in :
It is not hard to show that is the largest normal subgroup of contained in ,that is, andif and then .For this reason, some authors denote the core by rather than ,by analogy
with the notation for the normal closure
.
If ,then is said to be core-free.
If is of finite index in ,then is said to be normal-by-finite.
Let be the set of left cosets of in .By considering the action of on it can be shown thatthe quotient
(http://planetmath.org/QuotientGroup) embeds in the symmetric group
.A consequence of this is that if is of finite index in ,then is also of finite index in ,and divides (the factorial
of ).In particular, if a simple group
has a proper subgroup
of finite index ,then must be of finite order dividing ,as the core of the subgroup is trivial.It also follows thata group is virtually abelian if and only if it is abelian-by-finite,because the core of an abelian
subgroup of finite indexis a normal abelian subgroup of finite index(and the same argument applies if ‘abelian’ is replaced byany other property that is inherited by subgroups).
Title | core of a subgroup |
Canonical name | CoreOfASubgroup |
Date of creation | 2013-03-22 15:37:22 |
Last modified on | 2013-03-22 15:37:22 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 10 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 20A05 |
Synonym | core |
Synonym | normal core |
Synonym | normal interior |
Related topic | NormalClosure2 |
Defines | core-free |
Defines | corefree |
Defines | normal-by-finite |
Defines | core-free subgroup |
Defines | corefree subgroup |
Defines | normal-by-finite subgroup |