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.

270 PROF. BURNSIDE, ON A CLASS OF GROUPS OF FINITE ORDER.
other than those of order two. To deal first with the simplest case that presents
itself*, let
N= 2m,
where m is odd. Since no operation of order two is permutable with any operation of
odd order, G must contain m operations of order two which form a single conjugate
set. Let these be
Al, A2, ~......., Am.
If ArAs were an operation of order two, 1, A,, As, and ArAs, would constitute a
subgroup of G of order four. No such subgroup can exist, and therefore ArAs is an
operation of odd order. The m operations
A-rAi, ArA2,......, ArAm,
which are necessarily distinct, are therefore the m operations of odd order contained
in G. These m operations may similarly be expressed in the form
AlAr, A2Ar,......, AmAr;
and since
Ar. ArAs. A. = AsAr,
Ar transforins every operation of G, of odd order, into its inverse. Hence
ArAp. AqAr = AqAp = qAr. ArA,;
and this shews that every pair of operations of G, of odd order, are permutable. Hence
the m operations of G of odd order, including identity, constitute an Abelian group,
and this is a self-conjugate subgroup of G. Conversely, if H is any Abelian group of
odd order m, generated by the independent operations S, T,..., and if A is an
operation of order two such that
ASA = S-, ATA = T-1,...,
then A and H generate a group G of order 2m, whose only operations of even order
are those of order two.
When r is given, s can always be taken in just one way so that A,.A is any
given operation of G of odd order. Hence every operation of G of odd order can be
represented in the form ArAs in just m distinct ways. This property will be useful in
the sequel.
The groups thus arrived at are obviously analogous to the group (i) above.
2. Next let N= 2nm,
where m is odd and n is greater than one. The operations of order two contained
in G form one or more conjugate sets. Suppose first that they form more than one
such set; and let
A, A',...,
and B, B...,
* This first case is considered in my Theory of Groups of Finite Order, pp. 143 and 230.