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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 248.20068

**Autor: ** Eggleton, R.B.; Erdös, Paul

**Title: ** Two combinatorial problems in group theory. (In English)

**Source: ** Acta Arith. 21, 111-116 (1972).

**Review: ** Sequences of elements from (additive) abelian groups are studied. Conditions under which a nonempty subsequence has sum equal to the group identity 0 are established. For example, an n-sequence with exactly k distinct terms represents 0 if the group has order g \leq n+\binom{k}{2} and n \geq k\binom{k}{2}.

The least number f(k) of distinct partial sums is also considered, for the case of k-sequences of distinct elements such that no nonempty partial sum is equal to 0. For example, 2k-1 \leq f(k) \leq [ ^{1}/_{2} k^{2}]+1.

In this paper a sequence is a selection of members of a set, possibly with repetitions, in which order is not important; elements are members of sets, and terms are members of sequences.

Definition. Let * be a binary operation on a set A, and let S = (a_{i})^{n}_{i = 1} be a sequence of elements from A. S will be said to represent the element x in A if (i) x is a term in S, or (ii) there exist x,z in A such that x = y*z, and y and z are represented by disjoint subsequences of S. (Clearly this notion extends to general algebras.)

In particular, if < G,+> is an abelian group and S = (a_{i})^{n}_{i = 1} is a sequence of elements from G, then S represents x in G just if there exists a sequence E = (\epsilon_{i})^{n}_{i = 1} of elements from **{**0,1 **}**, not all 0, such that **sum** ^{n}_{i = 1} \epsilon_{i}a_{i} = x.

We resolve here some aspects of the following two related problems. (1) Under what circumstances does an n-sequence of elements from an abelian group represent the zero element? (2) If an n-sequence of distinct elements from an abelian group does not represent the zero element, how many elements does it represent?

**Classif.: ** * 20K99 Abelian groups

05A10 Combinatorial functions

05A20 Combinatorial inequalities

05-02 Research monographs (combinatorics)

00A07 Problem books

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