How Do I Understand <g> in Z/mZ?

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SUMMARY

The discussion centers on understanding the notation in the context of the group \mathbb{Z} / 7\mathbb{Z}, specifically when g = [5]. The user initially struggles with the concept of equivalence classes under modulo operations and the cyclic nature of the group. They clarify that represents the set of integer powers of g, and after some computation, they realize their confusion stemmed from fatigue rather than a lack of understanding. The user successfully computes when the group operation is addition, confirming their grasp of the concept.

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  • Familiarity with group theory concepts, particularly cyclic groups.
  • Knowledge of equivalence classes in the context of integers.
  • Basic skills in performing computations with powers in modular systems.
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  • Learn how to compute powers in \mathbb{Z} / n\mathbb{Z} for various values of n.
  • Explore the concept of generators in group theory.
  • Review the definition and properties of equivalence relations in mathematics.
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Students of abstract algebra, particularly those studying group theory, and anyone looking to deepen their understanding of modular arithmetic and cyclic groups.

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Hello. So today in class, we talked a bit about &lt;g&gt; as the set of all integer powers of g. Made enough sense. Then we did some examples in \mathbb{Z} / 7\mathbb{Z}, and I got a bit lost. I think this is more due to the fact that perhaps I don't quite grasp \bmod{n} as an equivalence class. So I know that the elements of \mathbb{Z} / 7\mathbb{Z} are \{ [0], [1], [2], [3], [4], [5], [6] \}, but we were looking for &lt;g&gt; where g = [5]. The idea was we were showing that \mathbb{Z} / 7\mathbb{Z} is cyclic. Since we were "brute-forcing" it so to speak, we wrote down elements in a certain order, which I have written down as \{ [0], [5], [3], [1], [6], [4], [2] \} and then it cycled back to [0].
So I suppose I'm having trouble doing computations in \mathbb{Z} / 7\mathbb{Z}, or \mathbb{Z} / n\mathbb{Z} in general. I mean, I know that if a \equiv b \bmod{n} then n \mid a-b, but in this case I'm not sure where to take it. Is it:
5^0 = 1 \equiv something \bmod{7}, then 5^1 = 5 \equiv something \bmod{7} etc?
Sorry if my question(s) is a bit hard to read; perhaps I didn't explain it as best I could. But any help would be appreciated.
 
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NEVER MIND!

Sorry, it's late, but I totally forgot the definition of $<g>$ when the group operation is addition. I computed it for that, and my results matched. I guess my problem is not with the computation, but rather when I need to take a break and go to sleep. I'm not sure I can close/delete this thread, so I'm posting this just in case.
 

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