# 1981-06 - ISRAEL JOURNAL OF MATHEMATICS. Vol 39, No. 3,...

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ISRAEL JOURNAL OF MATHEMATICS. Vol 39, No. 3, 1981 FINITE ABELIAN GROUP COHESION BY P. ERD&ANDB.SMITH ABSTRACT This paper studies the evenness of set arithmetic in a finite abelian group. Let G be a finite abelian group. We use # to denote cardinality. #G=p.For A,BCG let m(x,A,B)= #{(a,b): n+b=x, SEA, DEB}. For E C G let E' denote its complement. THEOREM. (Cohesion Equation). =x;G/m(x,E, -E)+m(x,E', -E')-m(x,E,-E')-m&E',-E)('. PROOF. Let r denote the dual group of G. Let 1 ifxEE, f(x) = -1 ifxEE’. Let p(x) = f( - x). The Cohesion Equation states Let j(r) = XxEGf(x)~(-x) for y E r. Then Received April 4, 1979 177

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178 P. ERDijS AND B. SMITH Israel J. Math. THEOREM 1. PROOF. Consider the right hand side of the Cohesion Equation. 2 /m&E, -E)+m(x,E’, -E’)-m(x,E, -E’)-m(x,E’, -E)l’ ZJm(O,E, -E)+m(O,E’, -E’)-m(O,E, -E’)-m(O,E’, -E)(*=p’. II. Let A > f. Let G be a finite group with no elements of order 2. Then (K depends only on A). The proof of Theorem II requires 3 Lemmas. For the remainder of the argument let # G = n + 1, and let there be no elements of order 2 in G. We consider all ways of writing G\(O) = E U F with #E = #F = n/2. Let a = (n - 1)/n. For x E G\(O) we see that cm/4 is the expected value of m(x, E, F), since (G\(O)) X (G\(O)) has cardinality nz and (G\(O)) -t (G\(O)) represents x( # 0) n - 1 times. When E is understood we use m(x) for m (x, E, F). Let where the summation is over s-tuples of integers, kl, * * *, k, satisfying: k, + . . * + k, = r. k, s k2 2 . . . 2 k, 2 1. k, = . . - = kj, ; kj,+l = - . - = kjl+h;. . . ; kj,+. ..+j,-,+I= .a* = kj,+. ..+j,. LEMMA ,zG (m(x) - anI4)P = I#0
Vol. 39, 1981 FINITE ABELIAN GROUP COHESION = P P )I( > \$ pn2(n - 2(l))(n - 2(2)) * * * (n - 2(p - 1)) 179 A (p, p - l)n’(n - 2(l)) * * . (n - 2(p - 2)) +; 0 p-2 A (p, p - 2)n*(n - 2(l)) . . . (n - 2(p - 3)) + -*.

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1981-06 - ISRAEL JOURNAL OF MATHEMATICS. Vol 39, No. 3,...

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