The Most Important Theorem in Finite Group Theory

Lagrange's Theorem is often described as the most important theorem in finite group theory. If G is a finite group and H is a subgroup of G, then the order of H divides the order of G. A corollary says that the order of any element of G divides the order of G. The converse of Lagrange's Theorem is false. For example, the alternating group A4, which has order 12, has no subgroup of order 6 even though 6 divides 12. The proof of Lagrange's Theorem is based on properties of cosets (left cosets aH and right cosets Ha). In particular, the collection of all left cosets of a subgroup H partitions the group G into subsets with the same cardinality (number of elements). The number of distinct left cosets is called the index of H in G and equals the order of G divided by the order of H.

#AbstractAlgebra #GroupTheory #LagrangeTheorem

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