Question
Asked Oct 17, 2019
Let G (Z,+) and let n be a positive integer. Let H nZ = {na : a E Z}. We know that
H is a subgroup of G. Describe the right cosets of H and G and find |G H.
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Let G (Z,+) and let n be a positive integer. Let H nZ = {na : a E Z}. We know that H is a subgroup of G. Describe the right cosets of H and G and find |G H.

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check_circleExpert Solution
Step 1

To determine (to list the elements of) the right cosets of the given subgroup 

Step 2

Now, Z is an abelian group , so the right cosets are the same as the left cosets. (every subgroup is normal). Let us first consider an example, say n =3 and this generalizes to all values of n

G Z, int egers under addition
H {3a: a E Z {..-6,-3,0,3,6,..
Any cos et is given by
x+H 3a+xa }
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G Z, int egers under addition H {3a: a E Z {..-6,-3,0,3,6,.. Any cos et is given by x+H 3a+xa }

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Step 3

The coset x+H is determined by the equival...

Any cos et is given by
x+H {3ax;ae Z
and x H yHxy(mod 3)
So,x
H is either H,1 H or 2+H
according as x 0,1, 2(mod 3)
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Any cos et is given by x+H {3ax;ae Z and x H yHxy(mod 3) So,x H is either H,1 H or 2+H according as x 0,1, 2(mod 3)

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