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

## Homework Statement

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

## 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?

SammyS
Staff Emeritus
Homework Helper
Gold Member

## Homework Statement

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

## 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?

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.

pasmith
Homework Helper

## Homework Statement

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

## 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.