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

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REGULAR SUBSTITUTION GROUPS

65

tutions into itself. It must therefore transform all of them into
complete sets of conjugates under G such that each of these
sets is composed of more than one substitution. As the total
number of these substitutions is prime to p, according to ~ 14,
Ex. 7, at least one of these sets of conjugates involves a
number m of substitutions, where mi is prime to p.
Each of these m substitutions is transformed into itself
by a subgroup of G whose order is g/m, where m> 1. Hence G
contains a subgroup whose order is divisible by pa. If this
subgroup is of order pa, our theorem is established. If it is not
of this order, we have reduced our problem to that of a smaller
group whose order is divisible by pt. In case Sylow's theorem
were not universally true it would clearly be possible to find
a smallest group G for which it would not be satisfied. As the
preceding considerations establish the fact that such a smallest
group does not exist, they constitute a proof of Sylow's theorem.
EXERCISES
1. If a group involves a subgroup whose order is one-half the order
of the group this subgroup is invariant.
2. If the order of a group is.pq, p and q being prime numbers and p>q,
this group is cyclic unless p-1 is divisible by q. In the latter case there
are exactly two groups of order pq.
3. Every simple group of composite order can be represented as a
non-regular transitive substitution group.
Suggestion: Consider the substitutions according to which any complete set of conjugate substitutions or subgroups are transformed under
the group.
4. There are exactly two abstract groups of order 4.*
Suggestion: Represent the possible groups as regular substitution
groups.
* The non-cyclic group of order 4 is known under various names. Among
these are the following: Axial group, four-group, fours group (Vierergruppe),
quadratic group, anharmonic group, and group of the general rectangle. For
the use of these terms, in order, the reader may consult the following: Pierpont,
Annals of Mathematics, vol. 1 (1900), p. 140; Bolza, American Journal of Mathematics, vol. 13 (1891), p. 75; B6cher, Introduction to Higher Algebra, 1907,
p. 87; Burnside, Theory of Groups of Finite Order, 1912, p. 444; Capelli, Istituzioni di analisi algebrica, 1909, p. 111; Miller, American Mathematical Monthly,
vol. 10 (1903), p. 217,