Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

142 PRIME-POWER GROUPS [CH. V
order p and the other having such a subgroup of order p2. When p =3
there is only one such group.
3. There are four distinct non-abelian groups of order p5, p>3, which
involve an abelian group of order p4, but do not contain any operator of
order p2. When p =3 there are only two such groups.
4. The number in can be so chosen that the number of the distinct
groups of order pm, p>2, which do not involve any operator of order p2
is greater than any given number.
5. Every intransitive Sylow subgroup of a symmetric group is the
direct product of its transitive constituents, and each of these transitive
constituents has a central of prime order.
6. If the degree of a symmetric group is n=klp"+k2p'-+...
+ka+l; kl, k2,.., ka,+ being positive integers less than p, then the
central of its Sylow subgroup of order pm is of order pkl+k2+ ' + a.