Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

272 PROF. BURNSIDE, ON A CLASS OF GROUPS OF FINITE ORDER.
than identity. It follows from an extension of Sylow's theorem that the number of
subgroups of order 2n contained in G must be of the form 2nk+ 1.
If K is the greatest subgroup of G which contains a subgroup H, of order 2n,
self-conjugately; then K must be a subgroup of the nature of those considered in the
preceding section, and its order must be 2~/, where /u is equal to or is a factor of
2 -1. Also no two operations of H can be conjugate in G unless they are conjugate
in K*. The 2n- operations of order two in K therefore form a single conjugate set;
and hence /u must be equal to 21 - 1. The order of G is therefore given by
T = (2nkc + 1) 2 (2n - 1).
That G must be a simple group is almost obvious. A self-conjugate subgroup of even
order must contain all the 2k + 1 subgroups of order 2n, since the operations of order
two form a single set. In such a subgroup the operations of order two must form a
single set, and therefore a subgroup of order 2n must be contained self-conjugately in
one of order 2nI(2n-1). Hence a self-conjugate subgroup of even order necessarily
coincides with G. If on the other hand G had a self-conjugate subgroup I of odd
order r, I would by the first section be Abelian and every operation of G of order
two would transform every operation of I into its inverse. This is impossible; for if A
and B were two permutable operations of order two in G which satisfy the condition,
then AB is an operation of order two which is permutable with every operation of I,
contrary to supposition. Hence G must be simple.
If A and B are any two non-permutable operations of order two in G, AB must
be an operation of odd order w, and A and B generate a dihedral group of order 2/L.
Hence G contains subgroups of the type considered in the first section. Let 2m, be
the greatest possible order of a subgroup of this type contained in G; and let If be a subgroup of G of order 2ml, and J1 the Abelian subgroup of order ml contained in 1,. Every
subgroup K of Ji is contained self-conjugately in I1; and, for the reason just given in
proving that G is simple, no two permutable operations of order two can transform K
into itself. Hence Il must be the greatest subgroup that contains K self-conjugately; as
otherwise 2mn would not be the greatest possible order for the subgroups of this type
contained in G.
Let pa be the highest power of a prime p which divides ml; and let K be a subgroup
of Ji of order pa. If pa is not the highest power of p which divides 1, then K would be
contained self-conjugatelyt in some subgroup of G of order pa+l. This has been proved
impossible. Hence ml and NV/ml are relatively prime.
Again no two subgroups conjugate to J, can contain a common operation other than
identity; for if they did Il would not be the greatest subgroup of its type contained
in G.
If Il and the subgroups conjugate to it do not exhaust all subgroups of G of order
2,u (/i odd), let I2 of order 2m2 (m, odd) be chosen among the remaining subgroups of G of
* Theory of Groups, p. 98. t Ibid. p. 65.