If $G$ a non cyclic abelian group, then $$G\cong \mathbb{Z}_{n_1}\times \mathbb{Z}_{n_2}\cdots\times \mathbb{Z}_{n_k}.$$ Can we give any presentation of this?
My attempt- $$G=\langle g_1g_2\cdots g_k:g_1^{n_1}=g_2^{n_1}=\cdots g_k^{n_k}=1, g_ig_j=g_jg_i;i,j=1,2,\cdots k \rangle,$$ where $\gcd(n_i,n_j)\neq 1, i\neq j, i,j=1,2\cdots k$.