MATHEMATICS: G. A. MILLER 
293 
suits in 'polygon-boden' on the other, every gradation of type of soil-flow 
may be found, and the combined results of their activities is a transportation 
of material as important as that of the streams and glaciers. 
All the field evidence tends to show that nivation andsolifluction, charac- 
teristic processes of disintegration and denudation under subarctic or arctic 
conditions, are of prime importance in the reduction of the high relief of 
northern Greenland. 
iEakin, H. M., Washington, U. S. Geol. Survey, Bull. 631, p. 76, 1916. 
2 Mathes, F. E., Washington, U. S. Geol. Survey, 21st Ann. Rep., p. 181, 1899-1900. 
3 Andersson, J. G., Chicago, J. Geol., Univ. Chicago, 14, 1906, p. 91. 
ON THE a-HOLOMORPHISMS OF A GROUP 
By G. A. Miller 
Department of Mathematics, University of Illinois 
Communicated by E. H. Moore, July 18, 1918 
The term a-holomorphism was introduced by J. W. Young to denote a 
simple isomorphism of a group G with itself which is characterized by the fact 
that each operator of G corresponds to its a th power. 1 A necessary and suf- 
ficient condition that an abelian group of order g admits an a-holomorphism 
is that a is prime to g, and J. W. Young proved in the article to which we re- 
ferred that when a non-abelian group admits an a-holomorphism the (a— \) th 
power of each of its operators is invariant under the group and the group ad- 
mits also an (a — l)-isomorphism. Moreover, these conditions are sufficient 
for the existence of an a-holomorphism. 
The object of the present note is to furnish a complete answer to the fol- 
lowing question : For what values of a is it possible to construct non-abelian 
groups which admit separately an a-holomorphism? It will be proved that no 
such group is possible when a is either 2 or 3, but that for every other posi- 
tive integral value of a there is an infinite system of non-abelian groups each 
of which admits an a-holomorphism. 
The fact that every group which admits a 2-holomorphism is abelian results 
directly from a theorem noted in the first paragraph of this article. If a non- 
abelian group G admits a 3-holomorphism we may represent two of its non- 
commutative operators by Si, s 2 , and note that as a result of this holomorphism 
the two dependent equations 
SiW = O1S2) 3 , Si 2 S 2 2 = O2S1) 2 
must be satisfied. Since $i 2 and s 2 2 are invariant under G it results directly 
from the latter equation that SiS 2 = s 2 Si, and hence the assumption that Si 
and s 2 are non-commutative led to a contradiction. That is, if a group admits 
a 3-holomorphism it must be abelian. 
