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. 273
this type so that m, is as great as possible; and let J2 be the Abelian subgroup of I, of
order m2. Then J, has no operation other than identity in common with J, or with any
subgroup conjugate to Ji; also no two subgroups conjugate to J2 have a common operation
other than identity, and m2 and N/m2 are relatively prime. All these statements may be
proved exactly as in the former case.
If the subgroups of G of order 2~L (,a odd) are still not exhausted, a subgroup I3 of order
2m3 containing an Abelian subgroup J, of order m3 may be chosen in the same way as
before; and the process may be continued till all subgroups of G of the type in question
are exhausted. Now JI is one of NV/2m conjugate subgroups and each contains n - 1
operations which enter into no other subgroup conjugate to J1 or to J2 or J3.... Hence
the subgroups conjugate to J, J2, J3,... contain
2n1 (ql)2 + 2m ((n2- 1) + (m3- ) +..
distinct operations other than identity. If 13 actually existed, this number would be equal
to or greater than N, which is impossible. Hence there can at most be only two sets of
conjugate subgroups such as Il and I,.
It was shewn in section 1 that each of the m - 1 operations of JI other than identity
can be represented in mr distinct ways as the product of two operations of order two.
Similarly each of the m, - 1 operations other than identity of J2, if it exists, can be represented
as the product of two operations of order two in m2 distinct ways. Moreover these and
the operations conjugate to them are the only ones which can be represented as the
product of two non-permutable operations of order two. Now G contains
(2Jc + 1) (2n-1)
operations of order two, and any one of these is permutable with exactly 2n- 1. Hence
the number of products of the form AB, where A and B are non-permutable operations
of order two and the sequence is essential, is
(2nk + 1) (2n - 1) 2nc (2n - 1) = Nk (2 - 1).
On the other hand as shewn above this number is
1T N
(mi -1)+- (m2- )
N
2or 2( 2-1
or 2 (mi-1)
according as Is actually exists or does not.
Hence if I2 does not exist
Ml = 2+ (2n - 1) + 1;
and at the same time m1 is a factor of
(2~1 + 1)(2' - 1).
VOL. XVIII. 35