Question
Asked Oct 25, 2019
Let G
(Z/18Z, +) be a cyclic group of order 18.
(1) Find a subgroup H of G with |H= 3
(2) What are the elements of G/H?
(3) Find a familiar group that is isomorphic to G/H.
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Let G (Z/18Z, +) be a cyclic group of order 18. (1) Find a subgroup H of G with |H= 3 (2) What are the elements of G/H? (3) Find a familiar group that is isomorphic to G/H.

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Step 1

The group G = (Z/18Z, +) be a cyclic group of order 18 so, the group G is similar to Z18.

 

Part (a):

 

A subgroup H of the group G such that |H|=3. Here 18 is divided by 3.

 

So, the subgroup of order 3 in a cyclic group of order 18 (Z18) is only one group which is Z3.

 

And similarly H can be written as:

H(6) (0,6,12) Z
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H(6) (0,6,12) Z

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Step 2

Part (b):

 

The elements of G/H:

Z,
H,1H,2H,3+ H,4+H,5+H
Н
{H,1 H, 2 H3 H,4 H,5H} forms a group under the operation,
which is isomorphic to Z
6
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Z, H,1H,2H,3+ H,4+H,5+H Н {H,1 H, 2 H3 H,4 H,5H} forms a group under the operation, which is isomorphic to Z 6

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Step 3

Z/H = {(0,6,12),(1,7,13),(2,8,14),(3,9,15),(4,10,16),(5,11,17)}

 

H={0,6,12}

 

H+1 = {1,7,13}

 

H+2 ={2,8,14}

&n...

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