  We redefine a multiplicative group structure on the set of equivalence
classes of rational sequences satisfying a fixed linear recurrence of degree
two, which was defined by R. R. Laxton in his paper "On groups of linear
recurrences I" published in Duke Math. 36, 721--736 (1969). In the article, he
also defined some natural subgroups of the group, and determined the structures
of their quotient groups. However, he did not study the whole group itself.
Nothing has been known about the structure of Laxton's whole group and its
interpretation. The aims of this paper are to redefine Laxton's group in a
natural way and determine the structure of the whole group itself, which
clarifies Laxton's results on the quotient groups. According to our formulation
by algebraic number theory method, we can simplify the proof of Laxton's
results. Our definition also gives a natural interpretation of Laxton's
results, and makes us possible to use the group to show various properties of
such sequences.
