The general linear group $GL_n(\mathbb{R})$

Closure. If $A,B$ are invertible, $AB$ is invertible with inverse $B^{-1}A^{-1}$.

Associativity. Matrix multiplication is associative.

Identity. $I_n$ is invertible and $I_nA=A=AI_n$.

Inverse. By definition, each $A\in GL_n$ has $A^{-1}\in GL_n$.

Non-abelian (for $n\ge2$). For example, $\begin{pmatrix}1&1\\0&1\end{pmatrix}\begin{pmatrix}1&0\\1&1\end{pmatrix}\neq\begin{pmatrix}1&0\\1&1\end{pmatrix}\begin{pmatrix}1&1\\0&1\end{pmatrix}$.

Conclusion. $GL_n(\mathbb{R})$ is a non-abelian group for $n\ge2$.