Groups, Cosets and Lagrange’s Theorem

Abstract

For the cryptosystems to be introduced in Chaps. 13 and 16 and for further study of RSA, we present some fundamental ideas in finite group theory, namely the concepts of a subgroup of a finite group and a coset of a subgroup, and Lagrange’s Theorem, a counting theorem involving a finite group, a subgroup and the cosets of that subgroup. Lagrange’s Theorem immediately implies Euler’s Theorem, a key result for the RSA cryptosystem. All of the groups used in this book are abelian, but for completeness, a concluding section introduces a non-abelian group with six elements to hint at the vast amount of group theory that is not discussed in this book.