Finding the number of elements in a cyclic group

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SUMMARY

The discussion focuses on determining the number of elements in a cyclic subgroup generated by the element 25 in the group Z30. The correct subgroup consists of six elements: {0, 5, 10, 15, 20, 25}. The operation in Z30 is addition modulo 30, and the confusion arose from miscalculating the generated elements. The participants clarified the generation process, demonstrating that repeated addition of 25 yields all six unique elements in the subgroup.

PREREQUISITES
  • Understanding of cyclic groups and subgroups
  • Familiarity with modular arithmetic
  • Knowledge of group theory terminology
  • Basic operations in Z30
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  • Study the properties of cyclic groups in abstract algebra
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  • Explore the concept of generators in group theory
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Mathematicians, students of abstract algebra, and anyone interested in group theory and modular arithmetic will benefit from this discussion.

hitmeoff
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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).
 
hitmeoff said:
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.
 

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