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I am beginner in group theory and I have learned cyclic groups yet. When I look at the books for definition, it is said that

Let $G$ be a cyclic group and let a be a generator of $G$ so that $$G= \langle a \rangle = {\{ a^n \mid n \in Z \}}$$

I know that the definition could have been like $G={\{ an \mid n \in Z \}}$ if the operation is addition. I also know that multiplication operation is used as general operation in groups theory, so the binary operations does not have to be multiplication.

So, can we generalize the definition for other binary operations except for multiplication and addition ? In other words, if a group is cyclic , then does the binary operation have to be multiplication or addition ? Is there any other cyclic group example whose binary operation is different from multiplication or addition such as random $*$? If so, how can we prove that they are abelian, because every cyclic group is abelian. It is easy to show that $a^n$ or $an$ is abelian, but what if the operation is $*$ for example.

Daniele Tampieri
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    Your question does not really make sense, this is just a matter of notation, here both imply the use of the binary notation in your group definition, except the second one implicitly tells you this is abelian (but both are abelian, except the first one can be an abelian subgroup of a larger non-abelian group) If you are more comfortable with it keep a strict notation like $\ast$. Also I think the "official" (maybe most canonical) definition of a cyclic group is a monogenic finite group – julio_es_sui_glace Mar 23 '24 at 15:51

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Powers of an element and the operation of a group are just notational; for example, as long as the axioms of group theory are satisfied by your use of notation, you could write, say,

$$a_n, n_a, \frac{a}{n}, \frac{n}{a}, (n)a, a^{(n)}, a\sim n, n\sim a, a\,\ddot\smile\, n . . .$$

or any other combination of symbols for "$a$ to the power of $n$".

This fact follows from the general philosophy of isomorphisms.

That said, for an abstract abelian group, we use addition and $na$ (for powers) most often.

Not to put too fine a point on it, but it's just a binary operation at the end of the day.

Every cyclic group is indeed abelian.

Shaun
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  • this is really interesting, so do you know any example of group whose operation is different from multiplication or addtion –  Mar 23 '24 at 16:08
  • See here, @juniorstudent. The operation $\triangle$ is used. – Shaun Mar 23 '24 at 16:09