How can an Infinity Group have the identity element e if e=e²=e³=...?

Question

From the definitions of ProofWiki whe have that an Infinite Cyclic Group G is a Cyclic Group, such that: Definition 2: $∀a∈G:∀m,n∈Z:m≠n⟹a^m≠a^n$.

But we know that if e is the identity element, e²=e³ and so on. What i'm missing here?

Answer 1

You are missing nothing. Instead of $\forall a\in G$, it should have been $\forall a\in G\setminus\{e\}$.

Answer 2

You need to assume $a\neq e$.