Find the order of the subgroups of (Z20,+) and the order of the subgroups. I kno

Question

Find the order of the subgroups of (Z20,+) and the order of the subgroups.

I know that the subgroups of (Z20,+) are <1>,<2>,<4>,<5>,<10>,<0>.

Now is the order as simple as o(<0>) = 1 ; o(<1>) = 20; o(<2>) = 10 ; o(<4>) = 5; o(<5>) = 4 ; o(<10>) = 2?

Or am I missing something. I am unsure of how exactly the order of the subgroup works. Do you only consider the subgroup itself or do you take the original group (Z20,+) into consideration like a normal order?

Explanation / Answer

It is as easy as you did it. The order of a subgroup is how many elements does the subgorup have.

For example, <4>={0,4,8,12,16}. hence it has 5 elements, so its order is 5.

Similarly, <5>={0,5,10,15}. It has 4 elements, so its order is 4.

If you have any other question, please comment and I will gladly answer. Please rate :0

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