TITLE: What does it mean "unique" for this author?
QUESTION [9 upvotes]: I'm studying Hungerford's Abstract Algebra book. I would like to know what the author means by "unique" in this theorem:

The orders count? I mean the element $g\in G$ such that $g=a_{i_1}a_{i_2}=a_{i_2}a_{i_1}$ is considered unique? 
I'm asking that because we know that if $G$ is an internal weak direct product of the family $\{N_i\mid i\in I\}$, we have $a_{i_k}a_{i_l}=a_{i_l}a_{i_k}$ for every $a_{i_k}\in N_{i_k}$ and $a_{i_l}\in N_{i_l}$.
Thanks in advance

REPLY [2 votes]: It's not written very clearly, but actually the order is intended not to matter because pairs of elements of two normal subgroups which intersect only at the identity in fact must commute, as you say:
Suppose $N_1$ and $N_2$ are normal subgroups of $G$ and $N_1 \cap N_2 = \{e\}$. Then let $a \in N_1$ and $b \in N_2$. Then $aba^{-1}b^{-1} = (aba^{-1})b^{-1} \in N_2$ but also $aba^{-1}b^{-1} = a(ba^{-1}b^{-1}) \in N_1$, so $aba^{-1}b^{-1} = e$, or in other words $ab = ba$.
Here's a nice write-up I found on internal weak direct products with a more clearly stated characterization which you may find helpful.