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. 275
by the cyclical subgroup of order 2-l1. But there are only just exactly 2n-.1
substitutions of order two in the 2+ 1 symbols which satisfy these conditions. Hence
for a given value of n the group, if it exists, is unique.
That such groups exist for all values of n is known*. In fact the system of
congruences
az+e
z'-_z+ (mod. 2),
yz + (m
where a, f, y, 8 are roots of the congruence
X2"-1 1, (mod. 2),
such that a S-y O 0, (mod. 2),
actually define such a group; and the permutations of the 2 +1 symbols
00,,, X2,...,X2"-1,
where X is a primitive root of
X2-1-_ 1, (mod. 2),
which are effected by the above system of congruences, actually represent it as a triplytransitive group of degree 2 + 1.
The set of groups thus arrived at are the analogues of group (iii) above.
Finally, every group of even order, which does not belong to one of the three sets
thus determined, must contain operations of even order other than operations of order
two.
NOTE.
Let H be an Abelian group of order 2n2 whose operations, except identity, are all
of order two; and suppose if possible that H admits two permutable isomorphisms of
prime order p one of which is not a power of the other, such that no operation of
order two is left unchanged by any isomorphism generated by the two. So far as a set
of p2 operations of H are concerned the two isomorphisms, being permutable, must have
the form
(A11AI..2 ) (A... Ap)) (A2A22... Ap). (AA... A),
and (An11A... Ap1) (A2A22... Ap)......(ApA... Ap);
A,1, A,2............. App,
Moore: "On a doubly-infinite series of simple groups," Chicago Congress Papers (1893); Burnside: "On
a class of groups defined by congruences," Proc. L. M. S. Vol. xxv. (1894).
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