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.

276 PROF. BURNSIDE, ON A CLASS OF GROUPS OF FINITE ORDER.
being the p2 operations. Moreover any cycle of an isomorphism generated by these two
has the form
(Ar, -Ar+, s+y...... A+(p-i), s-+(p-i)y),
the suffixes being reduced mod. p.
Since no operation of H except identity is left unchanged by any one of these
isomorphisms, the product of the p operations in any orle of the cycles must give the
identical operation.
Hence AlA21...... Ap, = 1,
AA22...... Ap, p = 1,
A11A23 1... Ap, p-l= 1.............................
AA2p.... Ap,2 =1;
and therefore on multiplication
A11 = 1,
or AJn =1.
The supposition made therefore leads to a contradiction. Hence if H admits a
group of isomorphisms of order pm, no one of which leaves any operations of H except
identity unchanged, this group has only a single subgroup of order p. It is therefore cyclical*.
If then pmn is the highest power of p which divides 27 - 1, the subgroup of order p1n
in the Abelian group of order 2n-1, considered above, is cyclical. Hence the Abelian
group is itself cyclical.
* Theory of Grozups, p. 73.