Download Factoring groups into subsets by Sandor Szabo, Arthur D. Sands PDF

By Sandor Szabo, Arthur D. Sands

Decomposing an abelian staff right into a direct sum of its subsets ends up in effects that may be utilized to quite a few parts, akin to quantity concept, geometry of tilings, coding thought, cryptography, graph thought, and Fourier research. Focusing generally on cyclic teams, Factoring teams into Subsets explores the factorization thought of abelian teams.

The ebook first exhibits how you can build new factorizations from previous ones. The authors then talk about nonperiodic and periodic factorizations, quasiperiodicity, and the factoring of periodic subsets. in addition they learn how tiling performs a major position in quantity thought. the subsequent numerous chapters disguise factorizations of limitless abelian teams; combinatorics, similar to Ramsey numbers, Latin squares, and intricate Hadamard matrices; and connections with codes, together with variable size codes, blunders correcting codes, and integer codes. the ultimate bankruptcy bargains with a number of classical difficulties of Fuchs.

Encompassing a number of the major components of the factorization thought, this booklet explores difficulties within which the underlying factored team is cyclic.

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5) We claim that (v) implies (i). 16 (on page 25), the sum A1 + · · · + An is direct. Since G= (Ai + bi ), bi ∈Bi the sum A1 + · · · + An is equal to G. Several times in the book we extended a factorization from a factorization of a subgroup or from a factorization of a factor group to the whole group. The essence of these procedures is summarized in the following lemma. 18 Let G be an abelian group and let H be a subgroup of G. Let H = B1 + · · · + Br be a factorization of H. Let C1 , . . , Cs be subsets of G such that © 2009 by Taylor & Francis Group, LLC 28 Applications of factoring cyclic groups G/H = (C1 + H)/H + · · · + (Cs + H)/H is a factorization of G/H.

Here A, B are normalized subsets and |A| = 2, |B| = 5. 8 (on page 18), the periods must belong to A and each of them must have order two. But the order of x + y is three. 8 (on page 18), its periods must have order 5. But G does not have any element of order 5. The next result shows that with suitable extra conditions, if A and B are not periodic, then neither is their sum A + B. 1 If A, B are normalized non-periodic sets of an abelian group and the sum A + B is direct, then A + B is a non-periodic set.

Xs+(p−1)p in P (x) are equal. This will turn out to be a useful result. Let G be a finite abelian group. To a subset A of G we assign the subset Kerχ. K= χ∈Ann(A) Here the intersection is taken over all characters χ of G that annihilate A. Note that Kerχ is a subgroup of G and consequently K is a subgroup of G. We call K the Corr´ adi subgroup of A. 1 Let G be a finite abelian group and let G = A + B be a factorization. If the Corr´ adi subgroup of A is not equal to {0}, then B is periodic. PROOF Let K be the Corr´adi subgroup of A and assume that K = {0}.

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