Show the group of units in Z_10 is a cyclic group of order 4

In summary, the conversation was about proving that the group of units in Z_10 is a cyclic group of order 4. The attempt at a solution involved showing that the elements 3 and 7 were isomorphic with Z_4, and therefore the group is cyclic. It was also noted that 1 cannot generate the whole group.
  • #1
HaLAA
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Homework Statement



Show that the group of units in Z_10 is a cyclic group of order 4

Homework Equations

The Attempt at a Solution


group of units in Z_10 = {1,3,7,9}

1 generates Z_4

3^0=1, 3^1=3, 3^2=9, 3^3= 7, 3^4= 1, this shows <3> isomorphic with Z_4

7^0=1 7^1= 7, 7^2= 9 7^3=3 7^4=1, this shows <7> isomorphic with Z_4

9^0=1 9^1=9 9^2 =1 9^3=9 9^4=1, this shows <9> doesn't isomorphic with Z_4

Did I do something wrong that I don't see this is a cyclic group of order 4?
 
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  • #2
HaLAA said:

Homework Statement



Show that the group of units in Z_10 is a cyclic group of order 4

Homework Equations

The Attempt at a Solution


group of units in Z_10 = {1,3,7,9}

1 generates Z_4

3^0=1, 3^1=3, 3^2=9, 3^3= 7, 3^4= 1, this shows <3> isomorphic with Z_4

7^0=1 7^1= 7, 7^2= 9 7^3=3 7^4=1, this shows <7> isomorphic with Z_4

9^0=1 9^1=9 9^2 =1 9^3=9 9^4=1, this shows <9> doesn't isomorphic with Z_4

Did I do something wrong that I don't see this is a cyclic group of order 4?
You didn't expect 1 to generate the whole group either, did you?
 
  • #3
SammyS said:
You didn't expect 1 to generate the whole group either, did you?
Right, 1 can't generate the whole. So there is only 3 and 7 isomorphic with Z_4.
 
  • #4
HaLAA said:

Homework Statement



Show that the group of units in Z_10 is a cyclic group of order 4

Homework Equations

The Attempt at a Solution


group of units in Z_10 = {1,3,7,9}

1 generates Z_4

3^0=1, 3^1=3, 3^2=9, 3^3= 7, 3^4= 1, this shows <3> isomorphic with Z_4

You're done here: your group contains four elements, and you've shown that it contains an element of order 4. Therefore it is cyclic.
 

1. What is a cyclic group?

A cyclic group is a mathematical concept that describes a group where every element can be generated by repeatedly applying a single element called a generator. In simpler terms, a cyclic group is a set of elements that can be "cycled" through by multiplying or combining with a single element.

2. What is Z10?

Z10 refers to the set of integers from 0 to 9, inclusive. It is often used to represent the numbers on a clock, where the numbers cycle through from 0 to 9 and then back to 0 again.

3. How do you show that the group of units in Z10 is cyclic?

To show that the group of units in Z10 is cyclic, we need to prove that there exists an element in the group that can generate all other elements in the group by repeatedly applying a group operation (in this case, multiplication).

4. What is the order of the cyclic group of units in Z10?

The order of a group is the number of elements in the group. In this case, the order of the cyclic group of units in Z10 is 4, meaning there are 4 elements in the group.

5. Can you provide an example of a generator for the cyclic group of units in Z10?

One example of a generator for this group is the number 3. By repeatedly multiplying 3 with itself, we can generate all other elements in the group: 32 = 9, 33 = 7, and 34 = 1 (since we are working in Z10, we take the remainder when dividing by 10). Thus, 3 is a generator for the cyclic group of units in Z10.

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