Let T denote the group of all nonsingular upper triaungular entries, i.e., the matrices of the form, [a,0,b,c] where a,b,c ∈ H H={[1,0,x,1]∈T} is a normal subgroup of T.

Question
Matrix transformations
Let T denote the group of all nonsingular upper triaungular entries, i.e., the matrices of the form, [a,0,b,c] where $$\displaystyle{a},{b},{c}∈{H}$$
$$\displaystyle{H}={\left\lbrace{\left[{1},{0},{x},{1}\right]}∈{T}\right\rbrace}$$ is a normal subgroup of T.

2021-01-16
$$\displaystyle{\left[{a},{0},{b},{c}\right]}^{{-{{1}}}}{\left[{1},{0},{x},{1}\right]}{\left[{a},{0},{b},{c}\right]}={\left[{1},{0},{c}\frac{{r}}{{a}},{1}\right]}∈{H}$$

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