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Exercise (What Is The Group Operation On The Integers)
If we want to consider $\mathbb{Z}$ as group, then which operation do we use, addition or multiplication? Why? Which is a group, is it $(\mathbb{Z},+)$ or is it $(\mathbb{Z},\cdot)$?
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The integers are a group under addition (the sum of two integers is an integer, the identity is 0, the inverse of $n$ is $-n$, and addition is associative as an axiom). For multiplication, the inverse of $2$, which is $1/2$, is not an integer so $\mathbb{Z}$ is not closed under inverses. There is no multiplicative inverse of 0 as $0x\neq 1$ for any integer $x$.