Groups and monoid in the set of Pythagorean triples (Q6542787)

In this paper, the author considers the groups and monoid in the set of Pythagorean triples.\N\NRecall that if \(x,y,\) and \(z\) are positive integers such that \(x^{2}+y^{2}=z^{2}\), then a triple \((x,y,z)\) is called a Pythagorean triple. In particular, if \(x,y,\) and \(z\) are coprime, the triple is called a primitive Pythagorean triple and if \(x,y,\) and \(z\) are rational, then it is called a rational Pythagorean triple.\N\NHe sets\N\[\NG=\{(a,b):a,b\in \mathbb{Z}, \gcd (|a|,|b|)=1\}\N\]\Nand sets\N\[\NE=\left\{ \left(\frac{a}{b},\frac{c}{d},\frac{e}{f}\right): \frac{a}{b},\frac{c}{d}, \frac{e}{f}\in \mathbb{Q},a,b,d,f\neq 0,\ \left(\frac{a}{b}\right)^{2}+ \left(\frac{c }{d}\right)^{2}=\left(\frac{e}{f}\right)^{2}\right\}\N\]\Nfor all \((a,b),(c,d),(e,f)\in G\). Then he proves\N\N\textbf{Lemma 1.} For any \(\left(\frac{a}{b},\frac{c}{d},\frac{e}{f}\right), \left(\frac{g}{h} ,\frac{i}{l},\frac{m}{n}\right)\in E\), the operation defined as\N\[\N\left(\frac{a}{b},\frac{c}{d},\frac{e}{f}\right)\cdot \left(\frac{g}{h},\frac{i}{l},\frac{m}{ n}\right)=\left( \frac{ag}{bh},\frac{cm}{dn}+\frac{ei}{fl},\frac{cm}{dn}+\frac{ei}{ fl}+ \left(\frac{e}{f}-\frac{c}{d}\right) \left(\frac{m}{n}-\frac{i}{l}\right)\right) \tag{1}\N\]\Nis a binary operation on \(E\).\N\NConsidering Lemma 1, he proves that\N\N\textbf{Theorem 3. }The set \(E\), together with the binary operation in (1) defined on \(E\) is a commutative infinite group with elements in \(\mathbb{Q}\).\N\NLater he sets\N\[\NM=\{(a,b,c):a,b,c\in \mathbb{Z},a^{2}+b^{2}=c^{2}\}\N\]\Nand defines a binary operation on \(M\) such that \N\[\N(a,b,c)\cdot (e,f,g)=(ae,bf+ce,bf+ce+(c-b)(f-e)). \tag{2}\N\]\NThen he proves that\N\N\textbf{Corollary 1.} The set \(M\) together with the binary operation in (2) defined on \(M\) is a commutative infinite monoid with elements in \(\mathbb{Z}\) and with elements in \(\mathbb{N}\) if \(a,b,c\in \mathbb{N}\).\N\NFor primitive Pythagorean triples, he sets\N\[\NP=\left\{ \left(\frac{a}{c},\frac{b}{c}\right): a,b,c\in \mathbb{Z},a,c\neq 0,\gcd (|a|,|b|,|c|)=1,a^{2}+b^{2}=c^{2}\right\}\N\]\Nand proves that\N\N\textbf{Theorem 4.} The set \(P\) together with the binary operation on \(P\), defined for all \(\left(\frac{a}{c},\frac{b}{c}\right), \left(\frac{d}{f},\frac{e}{f}\right)\in P\) as \(l:P\times P\rightarrow P\) such that\N\[\N\left(\frac{a}{c},\frac{b}{c}\right)\cdot \left(\frac{d}{f},\frac{e}{f}\right)=\left( \frac{ad}{ cf+eb},\frac{bf+ec}{cf+eb}\right)\N\]\N\N\[\N\left(\frac{a^{2}}{a^{2}},\frac{0}{a^{2}}\right) = \left(\frac{1}{1},\frac{0}{1}\right)\N\]\N\N\[\N\left( \frac{a^{2n}ad}{a^{2n}(cf+eb)},\frac{a^{2n}(bf+ec)}{a^{2n}(cf+eb)} \right) =\left( \frac{ad}{cf+eb},\frac{bf+ec}{cf+eb}\right)\N\]\Nfor all \(n\in \mathbb{N}\) is a commutative infinite group with elements in \( \mathbb{Q}\).\N\NHe also proves that\N\N\textbf{Theorem 5.} The set of all primitive Pythagorean triples is a commutative infinite group with elements in \(\mathbb{Z}\).\N\NFor all \((a,b,c),(d,e,f)\in M,\) he defines\N\[\Ni: M\rightarrow M\N\]\Nsuch that\N\[\Ni[(a,b,c)\cdot (d,e,f)]=(ad,bf+ec,cf+eb)\text{ }\N\]\Nif \((d,e,f)\neq (a,-b,c)\);\N\[\Ni[(a,b,c)\cdot (d,e,f)]=(1,0,1)\N\]\Nif \((d,e,f)=(a,-b,c)\) and\N\[\Ni[(a,b,c)\cdot (d,e,f)]=(a^{2n}ad,a^{2n}(bf+ec),a^{2n}(cf+eb))=(ad,bf+ec,cf+eb)\tag{3}\N\]\Nfor all \(n\in \mathbb{N}\). Then he proves\N\N\textbf{Corollary 2.} The set \(M\), together with the binary operation in (3) defined on \(M\) is a commutative infinite group with elements in \( \mathbb{Z}\).