Cauchy's Theorem: Showing that a Number is Greater than 1
Louis Halle Rowen, Uzi Vishne · Algebra · 2025
In Chapter 2 we saw that Lagrange&s;s theorem has the consequence that if https://www.w3.org/1998/Math/MathML" display="inline"> g ∈ G https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math3_1.tif "/> and https://www.w3.org/1998/Math/MathML" display="inline"> o ( g ) = m https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math3_2.tif "/> , then m divides https://www.w3.org/1998/Math/MathML" display="inline"> | G | https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math3_3.tif "/> . This raises the converse question: “If m divides https://www.w3.org/1998/Math/MathML" display="inline"> | G | https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math3_4.tif "/> then does https://www.w3.org/1998/Math/MathML" display="inline"> o ( g ) = m https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math3_5.tif "/> for suitable g in G ?” Our goal will be to prove this for m prime; for m not prime the claim is false, as evidenced by https://www.w3.org/1998/Math/MathML" display="inline"> Euler ( 8 ) = { 1 , 3 , 5 , 7 } https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math3_6.tif "/> , which has order 4 although each of its elements has order 2. In proving Lagrange&s;s theorem we examined the process of division. In studying the converse we shall learn how to count.