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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
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)) + -*.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 01/18/2012 for the course INFORMATIK 2011 taught by Professor Phanthuongcang during the Winter '11 term at Cornell.

Page1 / 9

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online