core of a subgroup
Let be a subgroup of a group .
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|
|Date of creation||2013-03-22 15:37:22|
|Last modified on||2013-03-22 15:37:22|
|Last modified by||yark (2760)|