Corollaries of Lagrange's Theorem in Group Theory
Lagrange's Theorem is so important in Group Theory from Abstract Algebra! Some important corollaries of Lagrange's Theorem include: 1) The index of a subgroup H in a finite group G is |G|/|H|. 2) If G is a finite group, the order of each element in G divides the order of G. 3) Every group of prime order p is isomorphic to ℤp (the group of integers under addition modulo p). 4) If G is a finite group and "a" is an element of G, then a^|G|=e (raising any group element to the power of the order of the group gives the identity element). 5) Fermat's Little Theorem: for every integer "a" and every prime p, a^p mod p equals a mod p.
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