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

CHAPTER V

GROUPS WHOSE ORDERS ARE POWERS OF PRIME NUMBERS
48. Introduction. It has been observed in ~ 11 that if
p" is the highest power of the prime p which divides the order
of a group G then G must involve at least one subgroup of order
pa, and if G involves more than one such subgroup, all the subgroups of this order (Sylow subgroups) form a complete set of
conjugates. These facts indicate that it is especially important
to know the fundamental properties of Sylow's groups; * that
is, of groups whose orders are powers of prime numbers. Fortunately all these Sylow groups have unusually interesting
properties in common and they offer more easy avenues of
penetration than the groups whose orders are arbitrary numbers.
A strong instrument of attack here, as well as in many other
places in group theory, is the concept of complete sets of conjugates. Each non-invariant operator of a non-abelian group
G of order pm belongs to a complete set of pa conjugates, since
such an operator is transformed into itself by all the operators
of a subgroup whose order is pI, F <m. Hence all the noninvariant operators of G occur in sets, such that each set involves
a power of p conjugate operators, and each non-invariant
operator occurs in one and in only one set. The total number
of the non-invariant operators must therefore be of the form
pk; and, as there are pm - operators besides the identity in
G, there must be an invariant operator of order p in G. This
* The groups whose orders are powers of prime numbers are also known as
primary groups. G. Frobenius and L. Stickelberger, Journal reine angew. Math.,
vol. 86 (1879), p. 219. They are sometimes called prime-power groups. In
view of the unusually large number of useful theorems in this field these groups
have been said to constitute the El Dorado of the theory of groups. Bulletin of
the American Mathematical Society, vol. 6 (1900), p. 393.
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