Do all finite groups contain an odd number of elements?

Question

Hello ive got a maybe weird and primitive question, which purely comes from my status as a beginner. Thinking about groups and their properties, with $\forall a \in G\,\,\, \exists a^{-1} \in G:aa^{-1}=e$ and the existence of a neutral element $e \in G$, doesnt this gives us the information, that any group, disregarding how large has a odd number of elements? And Cant we translate something into non-finite groups? This really interests me. Maybe someone can give me insight on how to look at this, and how to win or not win any information for non-finite sets.

Answer 1

No. For any $n\in\mathbb{N}$ the group $\mathbb{Z_n}=\{0,1,2,...,n-1\}$ where the operation is addition modulo $n$ is a group with $n$ elements.

Note that some elements satisfy $a=a^{-1}$, so $a$ and $a^{-1}$ are not always $2$ different elements. This is your mistake.