Let be a group of order 203. Prove that if is a normal subgroup of order 7 then . Prove that is abelian.
Note that . Now , so that . Moreover, since has prime order it is cyclic, hence abelian, and so , and thus (by Lagrange) divides 29. Now by Proposition 17. Since 29 and 6 are relatively prime, we have , hence . Thus .
Now is maximal. If , we have cyclic since (by Lagrange) is prime. Then is abelian, so that ; a contradiction. Thus , and we have abelian.