Question 2. The subgroup generated by S could have been defined a second way, as the set of all possible products of elements in S. Indeed, if g, and g2 are two elements in a subgroup of G then closure implies that the products (gi), (g2), (gig2)², (gig2)(g₁)³ (8182) (81)³(182) (g2)¹2, etc..... must also be in the subgroup. Define the closure of S to be the set: 5 = {ss sne Z,n ≥ 0 and s, ES, a, = ±1 for each 1 ≤ i ≤ n} and prove that (S) = S.

Question 2. The subgroup generated by S could have been defined a second way, as the set of all possible products of elements in S. Indeed, if g, and g2 are two elements in a subgroup of G then closure implies that the products (gi), (g2), (gig2)², (gig2)(g₁)³ (8182) (81)³(182) (g2)¹2, etc..... must also be in the subgroup. Define the closure of S to be the set: 5 = {ss sne Z,n ≥ 0 and s, ES, a, = ±1 for each 1 ≤ i ≤ n} and prove that (S) = S.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.1: Definition Of A Group

Problem 51E: Prove that the Cartesian product 24 is an abelian group with respect to the binary operation of...

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