We know that:
Theorem: if $G=\prod_{i=1}^n G_i$ be the direct product of groups $G_1,G_2,...,G_n$ then there exists normal subgroups $\bar{G_i}\cong G_i( i=1...n)$ of $G$ such that $G=\bar{G_1}\bar{G_2}...\bar{G_n}$ and $\forall i, 1\leq i\leq n; \bar{G_i}\cap(\bar{G_1}...\bar{G}_{i-1}\bar{G}_{i+1}...\bar{G_n})=\{1\}$
Maybe my question is so simple but: Could $\bar{G_i}$ be uniquely chosen, however, we don’t have this , stated in above theorem? Thanks.