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Let $*$ be defined on $\mathbb{Q}$ by letting $a*b=ab$

I know it needs to fulfill three axioms: Associativity, Identity, and Inverse. Associative is easy $a \cdot (b \cdot c) =(a \cdot b )\cdot c $.

The identity element is $1$, and $1 \cdot a = a \cdot 1 = a$

The Inverse $a'$ of $a$ somehow fails, and I don't get why, but I think it has something to do with $\mathbb{Q}$ containing $0$

Thank you in advance!

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    If $0$ has an inverse say $b$ then $0* b=b*0=1\implies 0.b=b.0=1\implies 0=1$ contradiction – 2017-02-10

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It absolutely has to do with 0. By definition of identity, ae=ea=a. By definition of inverse we know a' is the inverse of a if and only if a*a'=a'*a=e. Since this is multiplication the inverse of 0 must be the element of Q that you can multiply by 0 to get the identity, which we know is 1. But of course there is not even a real number which we can multiply by 0 to get 1. So since 0 has no inverse, < Q , * > is not a group.