Can someone teach me Lagrange's order theorem with specific numbers and cayley tables

Question

I am very new to mathematics and number theory. I understood some specifics of Groups, Rings, order of elements, order of group, Multiplicative group mod $N$, $U(N)$, etc., that is needed to understand RSA.

The last thing that I am stuck on is Lagrange's theorem.

All the aforementioned things I understood with the help of Cayley tables on other posts here. But for Lagrange, I can't find one. Everything I read is too abstract for me.

Can somebody make me understand the theorem with the help of specific numbers and Cayley tables please so that I can understand Euler-Fermat theorem and RSA in general?

Thank you :-)

Answer 1

I'm not sure I quite understand what you are looking for so I'll give you an example.

Lagrange's theorem just says that the order of any subgroup divides the order of a group.

For example, get a square of paper and consider $D_8$ the group isomorphic to the symmetries of a square. Now keep turning your square of paper around, this is a subgroup of $D_8$. It is either isomorphic to $C_4$ or $C_2$ depending on how big your turns are, generated by a rotation.

Lagrange's theorem simply implies that $|C_2|$ divides $|D_8|$ and $|C_4|$ divides $|D_8|$.

The theorem also gives us a better intuition for approaching new theorems, for example the orbit stabiliser theorem seems less bizarre having seen Lagrange's theorem. For a more whole understanding please see 'A Book of Abstract Algebra' by Charles C. Pinter specifically Chapter 3 pg 129 in the second edition proves this but the whole book is wonderful.

Answer 2

This is a reasonable question. The fact that the idea of "Cayley/multiplication tables" does not make Lagrange's theorem clear at all (except in some sort of laborious and unexplanatory gritty discussion in specific instances) is, in fact, evidence for the subtlety of Lagrange's theorem (although it arose very early in "group theory").

That is, viewing "the group operation" as simply some sort of look-up table with various rules absolutely does not explain why Lagrange's theorem should hold. The proof of it uses nothing about such look-up tables, but only considerably-more-abstract aspects of the notion of "group".

I think this situation is slightly similar to aspects of the question/fact about unique factorization of positive integers into primes. Namely, the existence follows for any particular not-too-large integer because we literally numerically factor it. Likewise the uniqueness follows in any particular case because there are only finitely-many conceivably alternatives to be checked, and none of them succeed. But all that does not hint at the mechanism that makes unique factorization provable in general. That is, numerical examples may be check-able to certify the conclusion of a theorem, without giving any good hint at a proof mechanism for that theorem.

Answer 3

Ims, the theorem is proved by noting that cosets of the subgroup both partition the parent group, and have the cardinality of the subgroup.

These two facts, fairly easy to establish, imply that the order of the subgroup divides that of the group.