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	\(\Bbb Z/7^* ={1,2,3,4,5,6} = \langle 3 \rangle \)

	\(\Bbb Z/6 = {0,1,2,3,4,5}\)

	Let \(\phi(3^{x})=x\)

	\(\phi(3^{x}3^{y}) = x+y=\phi(3^{x})\phi(3^{y})\)

	\(\phi\) is a homomorphism.

	Assume that \(\phi(3^{x})=\phi(3^{y})  \iff x=y\)

	\(x-y=0 \iff 3^{x-y}=e\) ; thus \(3^{x}=3^{y}\)

	\(\phi\) is injective

	Let \(K\in (\Bbb Z/7^*\) with \(0 \leq K < 7\)

\documentclass[10pt,a4paper]{article}
\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}
\begin{document}
	\(\Bbb Z/7^* ={1,2,3,4,5,6} = \langle 3 \rangle \)

	\(\Bbb Z/6 = {0,1,2,3,4,5}\)

	Let \(\phi(3^{x})=x\)

	\(\phi(3^{x}3^{y}) = x+y=\phi(3^{x})\phi(3^{y})\)

	\(\phi\) is a homomorphism.

	Assume that \(\phi(3^{x})=\phi(3^{y})  \iff x=y\)

	\(x-y=0 \iff 3^{x-y}=e\) ; thus \(3^{x}=3^{y}\)

	\(\phi\) is injective

	Let \(K\in \Bbb Z/6\) with \(0 \leq K < 6\)

\documentclass[10pt,a4paper]{article}
\usepackage[latin1]{inputenc}
\usepackage{amsmath}
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\usepackage{amssymb}
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\begin{document}
\(X=\)\(\begin{bmatrix} 1&a&b \\ 0&1&c \\ 0&0&1 \end{bmatrix}\)


\vspace{5mm}\vspace{10mm}

\(Y=\)\(\begin{bmatrix} 1&d&e \\ 0&1&f \\ 0&0&1 \end{bmatrix}\)\vspace{10mm}\vspace{10mm}

\(X\times Y =\begin{bmatrix} 1&a+d&b+af+e \\ 0&1&c+f \\ 0&0&1 \end{bmatrix}\) the product of upper triangular matrices is another upper triangular. thus \(H\) is closed under multiplication\vspace{10mm}\vspace{10mm}

\documentclass[10pt,a4paper]{article}
\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}
\begin{document}
\(X=\)\(\begin{bmatrix} 1&a&b \\ 0&1&c \\ 0&0&1 \end{bmatrix}\)


\vspace{5mm}\vspace{10mm}

\(Y=\)\(\begin{bmatrix} 1&d&e \\ 0&1&f \\ 0&0&1 \end{bmatrix}\)\vspace{10mm}\vspace{10mm}

\(X\times Y =\begin{bmatrix} 1&a+d&b+af+e \\ 0&1&c+f \\ 0&0&1 \end{bmatrix}\) the product of upper triangular matrices is another upper triangular. thus \(H\) is closed under multiplication\vspace{10mm}\vspace{10mm}