# Finding the number of elements in a cyclic group

## Main Question or Discussion Point

How do we go about finding the number of elements of a cyclic subgroup that's generated by an element in the main group. For example:

The subgroup Z30 generated by 25.

I would think this subgroup would be {0,1,5,25} but there's supposed to be 6 elements and not four. Whats going on?

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The operation in Z30 is addition, so your subgroup elements need to be generated by addition (for example, 25 + 25 = 20 (mod 30) must be in your subgroup).

HallsofIvy
Homework Helper
How do we go about finding the number of elements of a cyclic subgroup that's generated by an element in the main group. For example:

The subgroup Z30 generated by 25.

I would think this subgroup would be {0,1,5,25} but there's supposed to be 6 elements and not four. Whats going on?
How did you get that? 25+ 25= 50= 20 (mod 30). 25+ 25+ 25= 75= 15 (mod 30). 25+ 25+ 25+ 25= 100= 10 (mod 30). 25+ 25+ 25+ 25+ 25= 125= 5 (mod 30). 25+25+ 25+ 25+ 25+ 25= 150= 0 (mod 50). 25+25+ 25+ 25+ 25+ 25+ 25= 175= 25 (mod 30). Those are your 6 elements.

Thanks a lot guys. I guess I was just having a hard time reading my text and certain things weren't clear, I was confusing things.

You guys cleared it up.