# Tagged Questions

Use with the (group-theory) tag. A group is cyclic if it can be generated by a single element, $a$. Every element has the form $a^i$ for some $i\in\mathbb{Z}$, and so these groups are abelian.

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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,...
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### Groups of order $pq$ without using Sylow theorems

If $|G| = pq$, $p,q$ primes, $p \gt q, q \nmid p-1$, then how do I prove $G$ is cyclic without using Sylow's theorems?
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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.
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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 $G$,...
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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 ...
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### $(\mathbb{Z}/2^n \mathbb{Z})^*$ is not cyclic Group for $n\geq 3$

Question is to Prove that $(\mathbb{Z}/2^n \mathbb{Z})^*$ is not cyclic Group for $n\geq 3$. Hint : Find two subgroups of order $2$. I somehow feel that a cyclic group can not have two distinct ...
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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 ...
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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 ...
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### Give an example of a noncyclic Abelian group all of whose proper subgroups are cyclic.

I've tried but I could not find a noncyclic Abelian group all of whose proper subgroups are cyclic. please help me.
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### At most one subgroup of every order dividing $\lvert G\rvert$ implies $G$ cyclic [closed]

Suppose we have a finite group $G$ of finite order $n$. For every $d\mid n$, $G$ has at most one subgroup of order $d$. Show that $G$ is cyclic.
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### Homomorphism between cyclic groups

I have some confusion in relation to the following question. Let $\langle x\rangle = G$, $\langle y\rangle = H$ be finite cyclic groups of order $n$ and $m$ respectively. Let $f:G \mapsto H$ ...
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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 ...
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### Looking for a simple proof that groups of order $2p$ are up to isomorphism $\mathbb{Z}_{2p}$ and $D_{p}$ .

I'm looking for a simple proof that up to isomorphism every group of order 2p (p prime) is either $\mathbb{Z}_{2p}$ or $D_{p}$ (The Dihedral group of order 2p). I should note that by simple I mean ...
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### Showing that a group of order $21$ (with certain conditions) is cyclic

How can i show that if $o(G)=21$ and if $G$ has only one subgroup of order $3$ and only one subgroup of order $7$, then show that $G$ is cyclic.
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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 ...
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### Why $c(a_1 \ a_2 \dots \ a_k)c^{-1}=(c(a_1) c(a_2)… c(a_k))$?

We investigate on an arbitrary $a_i$ : $c(a_1 \ a_2 \dots \ a_k)c^{-1}(a_i)$. First step, $c(a_i)=a_k$. Second step, $(a_1 \ a_2 \dots \ a_k)(a_k)=a_{k+1}$, Third step, $c^{−1}(a_{k+1})=?$. Any ...
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### Finding homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{6}$.

Find all homomorphisms from $\mathbb Z_{12}$, the cyclic group of order $12$, to $\mathbb Z_6$. For each homomorphism $f\colon \mathbb Z_{12}\to \mathbb Z_6$, determine the kernel $\ker(f)$ and the ...
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### How to find all groups that have exactly 3 subgroups?

How to find all groups that have exactly 3 subgroups? Any group must have identity and itself as subgroups, so we just need to find all the groups that only have one proper subgroup. I think that ...
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### Can we conclude that this group is cyclic? [duplicate]

Let $G$ be a finite group. If, for each positive integer $m$, the number of solutions of the equation $x^m = e$ in $G$, where $e$ is the identity element, is at most $m$, then can we conclude that $G$ ...
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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 ...
### $n^2$ divides $\phi(a^n-1)$ whenever $n$ is even and $a>2$
My problem is to show that $n^2$ divides $\phi(a^n-1)$ whenever $n$ is even and $a>2$. I have thought a solution but it is quite long and tedious. I wonder if anyone has a nice and clear ...
Let $p$ be prime and let $\zeta_p$ be a primitive $p$th root of unity. Consider the quadratic subfield of $\mathbb{Q}(\zeta_p)$. For instance, for $p=5$ we get the quadratic subfield to be \$\mathbb{Q}(...