orbits of a normal subgroup are equal in size when the full group acts transitively
The following theorem proves that if a group acts transitively on a finite set, then any of the orbits of any normal subgroup are equal in size and the group acts transitively on them. We also derive an explicit formula for the size of each orbit and the number of orbits.
Let be a normal subgroup of , and assume acts transitively on the finite set . Let be the orbits of on . Then
permutes the transitively (i.e. for each , there is such that , and for each , there is such that ), and the all have the same cardinality.
If , then and .
Note first that if , and , then . For suppose also . Then since are in the same -orbit, we can choose such that . Then
since is normal in . Thus for each , there is such that .
Given , choose . Since is transitive on , we may choose such that . It follows from the above that .
To prove 1), given , choose such that and such that . But then so that and the subset relationships in the previous two paragraphs are actually set equality.
|Title||orbits of a normal subgroup are equal in size when the full group acts transitively|
|Date of creation||2013-03-22 17:17:56|
|Last modified on||2013-03-22 17:17:56|
|Last modified by||rm50 (10146)|