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

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1. May 17, 2015

### HaLAA

1. The problem statement, all variables and given/known data

Show that the group of units in Z_10 is a cyclic group of order 4
2. Relevant equations

3. 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?

2. May 17, 2015

### SammyS

Staff Emeritus
You didn't expect 1 to generate the whole group either, did you?

3. May 17, 2015

### HaLAA

Right, 1 can't generate the whole. So there is only 3 and 7 isomorphic with Z_4.

4. May 18, 2015

### pasmith

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.