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Asked Sep 17, 2019
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82

82. Let G be the set of all polynomi als of the form ax2 + bx + c with
coefficients from the set {0, 1, 2}. We can make G a group under
addition by adding the polynomi als in the usual way, except that
we use modulo 3 to combine the coefficients. With this operation,
prove that G is a group of order 27 that is not cyclic.
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82. Let G be the set of all polynomi als of the form ax2 + bx + c with coefficients from the set {0, 1, 2}. We can make G a group under addition by adding the polynomi als in the usual way, except that we use modulo 3 to combine the coefficients. With this operation, prove that G is a group of order 27 that is not cyclic.

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Expert Answer

Step 1

To prove that (1) G is a group or order 27 and (2) G is not a cyclic group

Step 2

(1) By very definition, G is isomorphic to the Cartesian product gro...

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G {ax +bx+c: a,b,c e Z, = {0,1,2}} As given,G is isomorphic to the product Z', Z {(a,b,c) a,b,ceZ,},which has 3 =27 elements So, G is a group with 27 elements as well

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