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.

PROF. BURNSIDE, ON A CLASS OF GROUPS OF FINITE ORDER. 271
be two distinct conjugate sets of operations of order two. The operation AB must
either be of order two or of odd order. If it were of odd order, /,, the subgroup
generated by A and B would be a dihedral subgroup of order 2/u; and in this subgroup A and B would be conjugate operations. Since A and B belong to distinct
conjugate sets in G, this is impossible. Hence AB is of order two, or in other words
A and B are permutable. Every operation of one of the two conjugate sets is therefore permutable with every operation of the other. The two conjugate sets therefore
generate two self-conjugate subgroups (not necessarily distinct) such that every operation
of the one is permutable with every operation of the other. The order of each of
these is divisible by two, and therefore the order of each must be a power of two;
as otherwise G would contain operations of order 2r (r odd). The two together will
generate a self-conjugate subgroup H' of order 2n'. If n' is less than n, there must
be one or more conjugate sets of operations of order two not contained in H'. Let
C, CC...,
be such a set. As before every operation of this set must be permutable with
every operation of H'. Hence finally G must contain a self-conjugate subgroup H of
order 2n. No operation of G is permutable with any operation of H except the
operations of H itself; and G is therefore a subgroup of the holomorph* of H. It
follows that G can be represented as a transitive group of degree 2n. Moreover, since
G contains no operations of even order except those of order two, the substitutions
of this transitive group must displace either all the symbols or all the symbols except
one. Hence m must be a factor of 2 - 1; and G contains 2n subgroups of order m
which have no common operations except identity. With the case at present under consideration may be combined that in which G has a self-conjugate subgroup of order
21, the 2' - 1 operations of order two belonging to which form a single conjugate set. In
this case m must be equal to 2 - 1.
We thus arrive at a second set of groups with the required property of order 2nm,
where m is equal to or is a factor of 2 - 1. They have a self-conjugate subgroup of
order 2n, and 2'1 conjugate subgroups of order m; the latter having no common operations
except identity. These are clearly analogous to group (ii) above.
3. Lastly there remains to be considered the case in which the operations of G
of order two form a single conjugate set, while G contains more than one subgroup
of order 2n.
If Il and H' are two subgroups of G of order 2n, and if I is the subgroup
common to H and H', then since H and H' are Abelian (their operations being all
of order two) every operation of I is permutable with every operation of the group
generated by H and H'. This group must have operations of odd order, since it contains
more than one subgroup of order 22. Hence I must consist of the identical operation
only; or in other words, no two subgroups of order 2n have common operations other
* Theory of Groups, p. 228.