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.

XI. On a Class of Groups of Finite Order. By Professor W. BURNSIDE.
[Received 30 September 1899.]
AMONG the groups of finite order that earliest present themselves, from some
points of view, to the student are the groups of rotations of the regular solids. An
admirable account of these from the purely geometrical stand-point is given in the
first chapter of Klein's Vorlesungen itber das Icosaëder. Of the six types included in
this set of groups there are three which, though quite unlike in other respects, have
a distinctive property in common. These are (i) the dihedral group of order 2n (n odd),
(ii) the tetrahedral group of order 12, and (iii) the icosahedral group of order 60.
They are defined abstractly by the relations:(i) A2=1, Bn=l, (AB)2 =, n odd;
(ii) A2=1, B3=1, (AB)3=1;
(iii) A2= 1, B3= 1, (AB)5= 1.
The order of each of these groups is even, while the only operations of even order
which they contain are operations of order two. While they have this property in
common they are otherwise of very distinct types.
The first has an Abelian (cyclical) self-conjugate subgroup, order n, which consists
of the totality of its operations of odd order. The second contains a self-conjugate subgroup of order four, this being the highest power of two which is a factor of the
order of the group. The third is a simple group containing five subgroups of order
twelve, each of which has a self-conjugate subgroup of order four. It can be represented as a triply-transitive substitution group of degree five.
I propose here to determine the groups of even order, which contain no operations
of even order other than operations of order two. The determination is exhaustive; and
it will -be seen that the groups in question arrange themselves in three quite different
sets of types of which the groups (i), (ii) and (iii), defined above, are representative.
1. Let G be a group of even order N, which contains no operations of even order