Let be a group of order 315 which has a normal Sylow 3-subgroup . Prove that and that is abelian.

Note that . The divisibility and congruence criteria of Sylow’s Theorem force and .

Note that , where is a group of order 6 or 48 by the comments at the end of §4.4. Hence , so that .

Consider now ; this is a group of order 35. By Sylow’s Theorem, has a unique Sylow 5-subgroup . Moreover, each element of has 3 preimages under the natural projection , for a total of elements in that map to .

Let be a Sylow 5-subgroup of . Now , so that . Note then that is a subgroup of order 5; hence . If , then at least elements of are mapped to by , a contradiction. Thus , so that is unique (as a Sylow 5-subgroup) and normal.

Now . Since , , so that and we have .

Now . By Lagrange’s Theorem, is either 45 or 315.

If , then is cyclic, so that is abelian. Hence , a contradiction. Thus , and we have . Thus is abelian.