property of infinite simple group
If a group is infinite and simple then it has no proper subgroups of finite index.
Let be an infinite simple group and . Then acts on the cosets of and this induces a homomorphism form to where . If has finite index in then is finite so maps homomorphically into the finite group . Thus the kernel of the homomorphism is non-trivial. As is simple, the kernel is . As contains the kernel, . ∎
This means that infinite simple groups do not act on finite sets so we cannot invoke clever arguments about the configuration of numbers. However linear representations may still apply. For example, for an infinite field is simple, infinite, and can be represented in through the exponential map of a Chevalley basis of the Lie algebra .
|Title||property of infinite simple group|
|Date of creation||2013-03-22 16:08:21|
|Last modified on||2013-03-22 16:08:21|
|Last modified by||Algeboy (12884)|