Observe that, by definition, , and that if , then , so that . Thus is a subgroup of . For , the subgroup is non-trivial, containing at least .
To illustrate an application of this concept we prove the following lemma.
We remark that , where denotes the center of .
Now let be a -group, i.e. a finite group of order , where is a prime and is a positive integer. Let . Summing over elements in distinct conjugacy classes, we have since the center consists precisely of the conjugacy classes of cardinality . But , so . However, is certainly non-empty, so we conclude that every -group has a non-trivial center.
The groups and , for any , are isomorphic.
|Date of creation||2013-03-22 12:35:01|
|Last modified on||2013-03-22 12:35:01|
|Last modified by||drini (3)|