Abstract:

Suppose G =Z_m1   Z_{m2    ^…}    Z_mr be an Abelian group of type (m₁, m₂, …, m_r) (Z_mi is a cyclic group of order m_i, 1≤ i ≤ r, m₁|m₂| …| m_r). Let S be a symmetrically closed set (S is symmetrically closed if 0 ∈ S =-S) and a generating set of G. If |S|>|G|/ρ, where ρ ≥lm_r /2 land m_r=e(G) denotes the least common multiple of the orders of all elements of  group G, then ρS=G. And if G=Z₂    Z_m is an Abelian group of type (2, m) (m ≥ 8), then t _m /2}(G)=0, which extends the related results of Klopsch and Lev^[1].

Key words: Cardinality, Generating sets, Symmetric closure