Easy# Group Definition: Examples involving Matrices

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A **group** is a nonempty set $G$ with a binary operation $(g,h)\mapsto g\circ h$ (so $g,h\in G$ implies $g\circ h\in G$), called its composition law:

(i) that is associative: $g\circ(h\circ k)=(g\circ h)\circ k$, all $g,h,k\in G$;

(ii) that has a neutral element $e\in G$, meaning $e\circ g=g\circ e=g$, all $g\in G$;

(iii) and such that every element $g\in G$ has an inverse $g^{-1}$, meaning $g\circ g^{-1}=g^{-1}\circ g=e$.

We say that $G$ is a group under $\circ$. Properties (i), (ii), (iii) above are often called the **group axioms**.

Let $n\ge 1$ be any integer. Which of the following are groups under the specified composition law?

Select **ALL** that apply.