While working on some group theory problems today a friend and I came up with the following result.
Proof. Let be the order of . Then the order of in divides both and . But . Hence , i.e., , i.e., .
Corollary 1. Let be a normal subgroup of a finite group such that . If such that divides , then .
Proof. Apply the lemma to the elements of .
Corollary 2. Let be a normal subgroup of a finite group such that . Then is the unique subgroup of of order .
Proof. Use Corollary 1.
Here is an example of the lemma in action.
Problem. Show that has no normal subgroup of order or .
Solution. If is a normal subgroup of of order , then . Hence every element of order or in must lie in . In particular, . By a result in the previous post, , a contradiction.
Likewise, if is a normal subgroup of of order , then must contain every 3-cycle; in particular, . Hence . But this has order 2, and , a contradiction.
More generally, we can prove the following.
Corollary 3. for has no non-trivial proper normal subgroup with .
Proof. Suppose otherwise and let divide . Then must contain all -cycles. So if is even then taking gives . If is odd, it contains the cycles and for some . Then has order 3. So contains all 3-cycles, i.e., . Since is maximal, either or , a contradiction.