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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.

A

The $n\times n$ matrices with real entries under matrix addition.

B

The $n\times n$ matrices with real entries under matrix multiplication.

C

The $n\times n$ matrices with real entries and non-zero determinant under matrix addition.

D

The $n\times n$ matrices with real entries and non-zero determinant under matrix multiplication.

E

The $n\times n$ matrices with real entries and determinant equal $1$ under matrix addition.

F

The $n\times n$ matrices with real entries and determinant equal $1$ under matrix multiplication.

G

The $n\times n$ matrices with real entries and determinant equal $-1$ under matrix addition.

H

The $n\times n$ matrices with real entries and determinant equal $-1$ under matrix multiplication.

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