You didn't expect 1 to generate the whole group either, did you?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?
Right, 1 can't generate the whole. So there is only 3 and 7 isomorphic with Z_4.SammyS said:You didn't expect 1 to generate the whole group either, did you?
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
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.
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.
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).
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.
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.