Abstract: The present article continues the classification of the
correlations of finite Desarguesian planes. The correlations of planes of
nonsquare order have been completely classified. We show in this paper that for
square orders, exactly as for nonsquare orders, a nonsingular matrix defining a correlation is
equivalent to an upper triangular matrix with at most five nonvanishing entries.
It has been convenient to discuss in a separate article the correlations of
planes of (even or odd) square order defined by diagonal matrices.
As the title indicates, this article is devoted to the case in
which the correlation of q an odd prime power, is defined by an upper triangular matrix with
four nonzero entries which is not reducible to a diagonal matrix. We show that
these correlations can have the following numbers of absolute points:
or or for n
odd;
or or for n
even.
We also discuss the equivalence classes into which these
correlations fall, as well as the configurations of their sets of absolute
points.
In a subsequent paper
we will consider the correlations of q a power of two, defined by upper triangular matrices with four
nonvanishing entries, while the concluding article will be concerned with
correlations of planes of (even or odd) square orders defined by upper
triangular matrices with five nonzero entries which are not reducible to
matrices with fewer such entries.
Keywords and phrases: companion automorphism, absolute set,
-equivalence, residue class, full secant, short secant.