Lagrange’s Theorem: Statement and Proof

lagrange s theorem : Statement and ProofPaul D. HumkeApril 5, 2002 AbstractLagrange s theorem is one of the central theorems of Abstract Algebra and it s Proof usesseveral important ideas. This is some good stuff to know!Before proving lagrange s theorem , we state and prove three a group with subgroupH, then there is a one to one correspondence betweenHand any coset a left coset ofHinG. Then there is ag Gsuch thatC=g :H Cbyf(x) =g one to , then asGhas cancellation,g x16=g x2. Hence,f(x1)6=f(x2). C, then sinceC=g H, there is anh Hsuch thaty=g h. It follows thatf(h) =yand asywas arbitrary,fis completes the Proof of Lemma a group with subgroupH, then the left coset relation,g1 g2if and only ifg1 H=g2 His an equivalence essence of this Proof is that is an equivalence relation because it is defined in termsofset equalityand equality for sets is an equivalence relation.

Lemma 1. If Gis a group with subgroup H, then there is a one to one correspondence between H and any coset of H. Proof. Let Cbe a left coset of Hin G. Then there is a g2Gsuch that C= g H.1 De ne f: H!Cby f(x) = gx. 1. fis one to one. If x 1 6= x 2, then as Ghas cancellation, gx 1 6= gx 2. Hence, f(x 1) 6= f(x 2). 2. fis onto.

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  Testament, Proof, Correspondence, Theorem, Lagrange, Lagrange s theorem, Statement and proof

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