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Math and science::Algebra::Aluffi

Groups. Lagrange's theorem.

Lagrange's theorem

Let G be a group, and let HG be a subgroup. Then:

|G|=|G/H||H|

There are three corollaries on the reverse. Can you remember them?

  1. About the order of g in G.
  2. When |G| is a prime.
  3. Fermat's little theorem.

|g| divides |G|

For any element gG, |g| divides |G|. In particular, |g| equals the order of the subgroup generated by g.

As a result, g|G|=eG for any element in a finite group G.

When |G| is a prime

If |G| is a prime p, then we must have GZ/pZ.

Fermat's little theorem

Let p be a prime, and let a be an integer. Then apamodp

Note on Fermat's little theorem

If one expects to see ap+1 instead of ap, then realize that if one considers the sequence e,a,a2,..., then the element a1 can be throught of as having already traversed two elements, e and a. So, to cover all p elements and by back to e we only need ap1, then another a brings us back to a.


Source

Aluffi p105