Algebra Solutions: A finite direct product is abelian if and only if each direct factor is abelian

Let $A$ and $B$ be groups. Prove that $A \times B$ is abelian if and only if $A$ and $B$ are abelian.

$(\Rightarrow)$ Suppose $a_1, a_2 \in A$ and $b_1, b_2 \in B$ . Then $(a_1 a_2, b_1 b_2) = (a_1, b_1) \cdot (a_2, b_2) = (a_2, b_2) \cdot (a_1, b_1) = (a_2 a_1, b_2 b_1)$ . Since two pairs are equal precisely when their corresponding entries are equal, we have $a_1 a_2 = a_2 a_1$ and $b_1 b_2 = b_2 b_1$ . Hence $A$ and $B$ are abelian.
$(\Leftarrow)$ Suppose $(a_1, b_1), (a_2, b_2) \in A \times B$ . Then we have $(a_1, b_1) \cdot (a_2, b_2) = (a_1 a_2, b_1 b_2) = (a_2 a_1, b_2 b_1) = (a_2, b_2) \cdot (a_1, b_1)$ . Hence $A \times B$ is abelian.

Algebra Solutions

A finite direct product is abelian if and only if each direct factor is abelian

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