Difference between revisions of "Lagrange's Theorem"
(statement, proof, and a few remarks) |
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By letting <math>K</math> be the [[trivial group | trivial subgroup]], we have | By letting <math>K</math> be the [[trivial group | trivial subgroup]], we have | ||
<cmath> |G| = (G:H) |H|. </cmath> | <cmath> |G| = (G:H) |H|. </cmath> | ||
− | In particular, if <math>G</math> is a [[finite]] group of [[order (group theory) | order]] <math> | + | In particular, if <math>G</math> is a [[finite]] group of [[order (group theory) | order]] <math>g</math> and <math>H</math> is a subgroup of <math>G</math> of order <math>h</math>, |
<cmath> g = (G:H) h, </cmath> | <cmath> g = (G:H) h, </cmath> | ||
so the index and order of <math>H</math> are [[divisor]]s of <math>g</math>. | so the index and order of <math>H</math> are [[divisor]]s of <math>g</math>. | ||
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* [[Quotient group]] | * [[Quotient group]] | ||
− | [[Category:Group theory]] | + | [[Category:Group theory]] [[Category: Theorems]] |
Revision as of 11:24, 9 April 2019
Lagrange's theorem is a result on the indices of cosets of a group.
Theorem. Let be a group, a subgroup of , and a subgroup of . Then
Proof. For any , note that ; thus each left coset mod is a subset of a left coset mod ; since each element of is in some left coset mod , it follows that the left cosets mod are unions of left cosets mod . Furthermore, the mapping induces a bijection from the left cosets mod contained in an arbitrary -coset to those contained in an arbitrary -coset . Thus each -coset is a union of -cosets, and the cardinality of the set of -cosets contained in an -coset is independent of the choice of the -coset. The theorem then follows.
By letting be the trivial subgroup, we have In particular, if is a finite group of order and is a subgroup of of order , so the index and order of are divisors of .