On the number of subsequence sums of a sequence in finite abelian groups related to the support of the sequence

Let G be a finite abelian group and S a sequence with elements of G. Let $|S|$ denote the length of S and k an integer with k∈ [1, |S|]. Let Σₖ(S) ⊂ G denote the set of group elements which can be expressed as a sum of a subsequence of S with length k. Let Σ(S)=∪ₖ₌₁|S|Σₖ(S) and Σ≥ k(S)=∪ₜ₌ₖ|S|Σₜ(S). It is known that if 0∈ Σ(S), then |Σ(S)|≥ |S|+|supp(S)|-1, where |supp(S)| denotes the number of distinct terms in S. In this paper, we study the above inequality. On one hand, we determine the sequence S with 0∉ Σ(S) such that |Σ(S)|= |S|+|supp(S)|-1. As a corollary, we disprove a conjecture of Gao, Grynkiewicz, and Xia. On the other hand, we generate the above inequality as if $|S|>k$ and 0∈ Σ≥ k(S)∪ Σ₁(S), then |Σ≥ k(S)|≥ |S|-k+|supp(S)|. Then among other results, we give an alternative proof of a conjecture of Hamidoune, which was first proved by Gao, Grynkiewicz, and Xia.