Sequenceable groups: a survey
A. Donald Keedwell · Cambridge University Press eBooks · 1981
A finite group (G,·) of order n is said to be sequenceable if its elements can be arranged in a sequence a 0 , a 1 , a 2 ,…, a n-1 in such a way that the partial products b 0 = a 0 , b 1 = a 0 a 1 , b 2 = a 0 a 1 a 2 ,…, b n-1 = a 0 a 1 … a n-1 are all distinct and so are again the elements of G. It is immediately evident that for this to be possible, the first element a 0 must be equal to the identity element e of G. Sequenceable groups arise in connection with the construction of so-called complete latin squares. A latin square on n symbols is called row complete if each of the n(n-l) ordered pairs of distinct symbols occurs in adjacent positions (cells) in exactly one row of the latin square. Since there are n-l pairs of adjacent cells in each row of the square, we get an exact match between the ordered pairs and the places in which they may occur. An example is given in Figure 1. There is an analogous definition of column completeness . A latin square which is both row complete and column complete is called complete . The square given in Figure 1 is a complete latin square. A practical application of row complete latin squares of small size is to the statistical design of sequential experiments in which several treatments are to be administered in succession to a number of different subjects.