Which of the following is false? Choose one answer. a. S_3 isgenerated by the tr

Question

Which of the following is false? Choose one answer. a. S_3 isgenerated by the transpositions (12) and (13)
b. S_3 isgenerated by the 3-cycles (123) and (132)
c. the integersunder addition are generated by 1 and -1
d. themultiplicative group {1,i,-1,-i} is generated by -i Which of the following is false? Choose one answer. a. S_3 isgenerated by the transpositions (12) and (13)
b. S_3 isgenerated by the 3-cycles (123) and (132)
c. the integersunder addition are generated by 1 and -1
d. themultiplicative group {1,i,-1,-i} is generated by -i Choose one answer. a. S_3 isgenerated by the transpositions (12) and (13)
b. S_3 isgenerated by the 3-cycles (123) and (132)
c. the integersunder addition are generated by 1 and -1
d. themultiplicative group {1,i,-1,-i} is generated by -i Choose one answer. a. S_3 isgenerated by the transpositions (12) and (13)
b. S_3 isgenerated by the 3-cycles (123) and (132)
c. the integersunder addition are generated by 1 and -1
d. themultiplicative group {1,i,-1,-i} is generated by -i

Explanation / Answer

(a) suppose a = (12) then a2 = e the identityand a3 = a , a4 = e , ---- similarly if b = (13) then b2 = e , b3 =b , b4 = e , b5 = b and so on. keeping these things in view, <a,b> = { a, b , e} notequal to S3 a, b are not the generators of S3 (b) suppose a =(123) , b=(132) since S3 is anon abelian group, these two members can generate S3. (c) when a is an integer , a+1 is also an integer and 1+(-1)produces the identity and so on. so, 1 and -1 are the generators of integers. (d) both i and - i are the generators of {1,-1,i,-i}. (c) when a is an integer , a+1 is also an integer and 1+(-1)produces the identity and so on. so, 1 and -1 are the generators of integers. (d) both i and - i are the generators of {1,-1,i,-i}.

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