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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.


Solution:

($\Rightarrow$) 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.

Linearity

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