On Certain Zero-sum Problems in Finite Abelian Groups
Ravindranathan Thangadurai · Hindustan Book Agency · 2002
Let G be a finite additive Abelian group. By the structure theorem of finite Abelian groups, we know, there exists integers 1 < n1|n2|⋯|n r such that ℤ n1 ⊕ ℤ n2 ⊕…⊕ ℤ nr . $$G \cong {\mathbb{Z}_{n1}} \oplus {\mathbb{Z}_{n2}} \oplus \ldots \oplus {\mathbb{Z}_{nr}}.$$ Then, r = r(G) is called the rank of G and n r = exp(G) is called the exponent of G. By a sequence S = {a1, a2, ⋯, an} in G, we mean ai ∊ G and ai’s are not necessarily distinct. Here, n is called the length of S. We define ΣS:={ t∈G:t= a i 1 + a i 2 +…+ a i l , i 1 , i 2 ,…, i l distinct, 1≤l≤n }. $$\Sigma S: = \left\{ {t \in G:t = {a_{{i_1}}} + {a_{{i_2}}} + \ldots + {a_{{i_l}}},{i_1},{i_2}, \ldots ,{i_l}\quad {\text{distinct,}}\quad 1 \leqslant l \leqslant n} \right\}.$$