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, and
if 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 that
the 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 that
a group is virtually abelian if and only if it is abelian-by-finite,
because the core of an abelian
subgroup of finite index
is a normal abelian subgroup of finite index
(and the same argument applies if ‘abelian’ is replaced by
any 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 |