By S. M. Gersten, John R. Stallings

Group conception and topology are heavily similar. The zone in their interplay, combining the logical readability of algebra with the depths of geometric instinct, is the topic of Combinatorial workforce thought and Topology. The paintings contains papers from a convention held in July 1984 at Alta resort, Utah.

Contributors to the ebook comprise Roger Alperin, Hyman Bass, Max Benson, Joan S. Birman, Andrew J. Casson, Marshall Cohen, Donald J. Collins, Robert Craggs, Michael Dyer, Beno Eckmann, Stephen M. Gersten, Jane Gilman, Robert H. Gilman, Narain D. Gupta, John Hempel, James Howie, Roger Lyndon, Martin Lustig, Lee P. Neuwirth, Andrew J. Nicas, N. Patterson, John G. Ratcliffe, Frank Rimlinger, Caroline sequence, John R. Stallings, C. W. Stark, and A. Royce Wolf.

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N 2, j = 1, ... , t + 2, corresponding to the set -Iffnt in the following way. First we construct the n2 x 2 submatrix consisting of the first two standard columns, whose rows are the lexicographically ordered pairs of numbers 1, ... , n. , t + 2. 6). Indeed, the equality (cil, CO = (c jl, c j2), i * j, contradicts the property of the rows of the standard columns. , t + 2, contradict that property of Latin squares which states that there are no repetitions of elements in the rows and in the columns.

6). We now consider sets of t mutually orthogonal Latin squares of order n with substitutions acting on the set { 1, ... , n}. The following lemma will be used later in this section. 6), that is, the matrix can be constructed for a given set ant and vice versa. Proof Suppose we are given a set ant consisting oft pairwise orthogonal Latin squares L;,1>, ... , Lu). ,n 2, j = 1, ... , t + 2, corresponding to the set -Iffnt in the following way. First we construct the n2 x 2 submatrix consisting of the first two standard columns, whose rows are the lexicographically ordered pairs of numbers 1, ...

Thus n = 4p, where p is a natural number, and the theorem is proved. We now define the Kronecker product of two square matrices. If A = II aiJ II, i, j = 1,... , n, and B = II bil II, i, j = 1, ... , m, then the Kronecker product A ® B of the matrices A and B is the mn x mn square matrix A®B= a11B a21B a12B a22B ... a2nB ... annB aj,B , an1B an2B In other words, C=A B = II cs;,s; II, i, j = 1, ... , mn, where csis; = ai1l1 bi2h, and s1, s2, ... , s,nn are pairs si = (il, jl ), sj = (j1, j2) of the form (i, j), with i = I,-, n, j = 1, ...

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