A group is a set G together with a multiplication, which

satisfies three properties. A multiplication is a binary operator

$$ : G $\backslash$times G $->$ G

which takes two inputs from G and returns one output from G$.$ The three properties that the

multiplication $$ is required to satisfy are the following:

$(1)$ for all elements g$,$ h$,$ k in G of G$,$ we have $($g $$ h$)$ k $=$ g $($h $$ k$),$

$(2)$ there exists an element $1$G in G such that for all g in G$,$ we have g $1$G $=1$G $$ g $=$ g$,$

$(3)$ for all elements g in G$,$ there exists g$1$ in G such that g $$ g$1=$ g$1$ g $=1$G$.$

This structure is absolutely central in mathematics since its introduction in the $19$th century.

In this exercise, your task will be to show that the following set of matrices is a group under

matrix multiplication

G :$=$

$$

$$

$$

$$

$$

$1$ a b

$01$ c

$001$

$$

$$ a$,$ b$,$ c in R

$$

$.$