On the symmetric difference of two sets in a group
type: Item
instance of: scholarly article
title: On the symmetric difference of two sets in a group
zbMATH Open document ID: 0597.05012
DOI: 10.1016/S0195-6698(86)80016-6
author: John E. Olson
published in: European Journal of Combinatorics
publication date: 1986-01-01T00:00:00Z
review text: In the proof of certain combinatorial theorems on sums of sets in a general group the results on symmetric differences of two sets play an important role. The first main result of the paper is as follows: Suppose X and Y are finite subsets of a group G, H and K are subgroups, \(X+H=X\), \(Y+K=Y\), \(X+K\neq X\), and \(Y+H\neq Y\). Then \(| X\setminus Y| +| Y\setminus X| \geq | H| +| K| -2| H\cap K|\). This removes the hypothesis that H, K are normal subgroups in the known results. This removal leads, however, to two possible modifications of the hypothesis. The second main result of the paper contains the alternative in which the subgroups H, K fix the sets X, Y on the opposite sides, i.e. \(H+X=X\) and \(Y+K=Y\). Further, in an erlier paper [J. Number Theory 18, 110-120 (1984; Zbl 0524.10043)] the author proved that if \(C=A+B\), where A, B are finite subsets in a group G, then there is a subset S of C such that \(| S| \geq | A| +| B| - | H|\). In the present paper he proves that among the sets S satisfying this conclusion there is a unique maximal one that include all the others. In the last part of the paper a conjecture of \textit{G. T. Diderrich} [Proc. Am. Math. Soc. 38, 443-451 (1973; Zbl 0266.20041)] is disproved by constructing a pair of sets A, B in a group G of order 144 satisfying \(| A+B| <| A| +| B| -1\), but no one of the relations \(g+A+B=A+B\), \(A+g+B\subseteq A+B\), \(A+B+g=A+B\) has a solution \(g\neq 0\) in G.
Mathematics Subject Classification ID: 05B10
zbMATH DE Number: 3961624
zbMATH Keywords: Kneser's theorem
zbMATH Keywords: additions theorems
zbMATH Keywords: Diderrich's conjecture
zbMATH Keywords: symmetric differences
reviewed by: Štefan Porubský
MaRDI profile type: Publication
cites work: On Kneser's Addition Theorem in Groups
cites work: Q3235825
cites work: Abschätzung der asymptotischen Dichte von Summenmengen
cites work: Q5334400
cites work: On the sum of two sets in a group
full work available at URL: s0195-6698(86)80016-6
OpenAlex ID: W2023042611
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description: scientific article; zbMATH DE number 3961624