Questions tagged [cyclic-groups]

Use with the (group-theory) tag. A group is cyclic if it can be generated by a single element. That is to say, every element in a cyclic group can be written as some specified element to a power.

A group $G$ is cyclic if it can be generated by a single element $a$. This means that any element of a cyclic group has the form $a^n$ for some integer $n$. Notationally, we often write that $G$ is isomorphic to $\langle a \rangle$. Since

$$a^na^m=a^{n+m}=a^{m+n}=a^ma^n\,,$$

cyclic groups must be abelian. Note though that the generator is not necessarily unique: for example the cyclic group $\mathbf{Z}/7\mathbf{Z}$, consisting of the elements $\{0,1,\dotsc,6\}$ and equipped with the operation of addition modulo $7$, can be generated by any of its non-identity elements.

Cyclic groups are completely classified. Up to isomorphism, $\mathbf{Z}$ equipped with addition is the only infinite cyclic group. Every finite cyclic group is isomorphic to a group of the form $\mathbf{Z}/n\mathbf{Z}$, a quotient of the integers under addition modulo $n$.

Cyclic groups are incredibly useful in describing the structure of finite abelian groups. By the classification theorem of finite abelian groups, every finite abelian group is isomorphic to a direct sum of cyclic groups, each having order a power of a prime.

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If $G/Z(G)$ is cyclic, then $G$ is abelian

Continuing my work through Dummit & Foote's "Abstract Algebra", 3.1.36 asks the following (which is exactly the same as exercise 5 in this related MSE answer): Prove that if $G/Z(G)$ is cyclic, then $G$ is abelian. [If $G/Z(G)$ is cyclic with…
Altar Ego
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For what $n$ is $U_n$ cyclic?

When can we say a multiplicative group of integers modulo $n$, i.e., $U_n$ is cyclic? $$U_n=\{a \in\mathbb Z_n \mid \gcd(a,n)=1 \}$$ I searched the internet but did not get a clear idea.
Sankha
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How to find a generator of a cyclic group?

A cyclic group is a group that is generated by a single element. That means that there exists an element $g$, say, such that every other element of the group can be written as a power of $g$. This element $g$ is the generator of the group. Is that…
user3543192
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Subgroups of a cyclic group and their order.

Lemma $1.92$ in Rotman's textbook (Advanced Modern Algebra, second edition) states, Let $G = \langle a \rangle$ be a cyclic group. (i) Every subgroup $S$ of $G$ is cyclic. (ii) If $|G|=n$, then $G$ has a unique subgroup of order $d$ for each…
user58289
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Order of automorphism group of cyclic group

Let $G$ be a cyclic group of order $m$. What is the order of $\text{Aut}(G)$? I want to know the proof as well (elementary if possible). I would still accept the proof if one answers with $m = p$, a prime. Or on top of that, I would accept the…
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Show that every group of prime order is cyclic

Show that every group of prime order is cyclic. I was given this problem for homework and I am not sure where to start. I know a solution using Lagrange's theorem, but we have not proven Lagrange's theorem yet, actually our teacher hasn't even…
user2467
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Are cyclic groups always abelian?

If a group $C$ is cyclic, is it also abelian (commutative)? If so, is it possible to give an “easy” explanation of why this is? Thanks in advance!
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A subgroup of a cyclic group is cyclic - Understanding Proof

I'm having some trouble understanding the proof of the following theorem A subgroup of a cyclic group is cyclic I will list each step of the proof in my textbook and indicate the places that I'm confused and hopefully somewhere out there can clear…
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Product of two cyclic groups is cyclic iff their orders are co-prime

Say you have two groups $G = \langle g \rangle$ with order $n$ and $H = \langle h \rangle$ with order $m$. Then the product $G \times H$ is a cyclic group if and only if $\gcd(n,m)=1$. I can't seem to figure out how to start proving this. I have…
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Finite groups with exactly one maximal subgroup

I was recently reading a proof in which the following property is used (and left as an exercise that I could not prove so far). Here is exactly how it is stated. Let $G$ be a finite group. Suppose it has exactly one maximal subgroup. Then $G$ is…
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Proof that all abelian simple groups are cyclic groups of prime order

Just wanted some feedback to ensure I did not make any mistakes with this proof. Thanks! Since $G$ is abelian, every subgroup is normal. Since $G$ is simple, the only subgroups of $G$ are $1$ and $G$, and $|G| > 1$, so for some $x\in G$ we have…
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Why must a field whose a group of units is cyclic be finite?

Let $F$ be a field and $F^\times$ be its group of units. If $F^\times$ is cyclic, then show that $F$ is finite. I'm a bit stuck. I know that I can represent $F^\times = \langle u \rangle$ for some $u \in F^\times$ and that we must have that…
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If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.

Exercise 11, page 45 from Hungerford's book Algebra. If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgroup of $H$ is normal in $G$. I am trying to show that $a^{-1}Ka\subset K$, but I got stuck. What I am supposed to…
spohreis
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All elements in $\mathbb{Z}/n\mathbb{Z}$ are representable as sum of a square and a cube?

Is it true that all elements in $\mathbb{Z}/n\mathbb{Z}$ are representable as the sum of a square and a cube? Example: ($n=7$) $0 \equiv 0^2+0^3 \left( \text{mod } 7 \right)$ $1 \equiv 1^2+0^3 \left( \text{mod } 7 \right)$ $2 \equiv 1^2+1^3 \left(…
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Prove that $\mathbb{R^*}$, the set of all real numbers except $0$, is not a cyclic group

Prove that $\mathbb{R^*}$ is not a cyclic group. (Here $\mathbb{R^*}$ means all the elements of $\mathbb{R}$ except $0$.) I know from the definition of a cyclic group that a group is cyclic if it is generated by a single element. I was thinking…
Student
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