# 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.

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### Find the solutions of $a^5=32$ in $Z^{*}_{2081}$ [duplicate]

Find the solutions of $a^5=32$ in $Z^{*}_{2081}$ As you directly see, the first solution is $2$. To find other roots, I tried to make use of the theorem such that if $x^d=e$ and $d$ divides the order ...
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### Determine the number of group homomorphisms $f:\mathbb Z_{63}\to\mathbb Z_{147}$ with $|image(f)|=7$.

Solution is apparently 6. Current attempt, although, I am not sure whether this the right approach: Let $f(x):=x$. We require $f(63)=63x\equiv 0 \mod 147$; that is, $63x=147k\iff 7\mid x$. So we have ...
• 21
1 vote
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### Understanding Why a Product of Cyclic Groups with Non-Coprime Orders is not Cyclic

Let $G_1, G_2, \ldots, G_t$ be finite cyclic groups, and define $G = G_1 \times G_2 \times \ldots \times G_t$. Let $n_j = |G_j|$, such that $|G| = \prod_{j=1}^{t} n_j$. For part (a) of the problem, we ...
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### Geometric Algebra: No other point groups besides $H_p$ and $C_p$ in two dimensions

I have been reading the paper "Point Groups and Space Groups in Geometric Algebra" by David Hestenes as part of my introduction seminar to GA. On page 5 the remark is made, that to prove ...
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### Why is $\langle 3 \rangle$ a generator of $(\mathbb{Z}_7^*, \cdot)$?

This link here shows that $\langle 3 \rangle$ is a generator for the given group by brute force method, that is trial and error. I was curious as to how to justify this using the theorem. According to ...
1 vote
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### Multiplicative orders modulo divisors of the modulus

Is there a known description of the set of multiplicative orders of a fixed unit $a$ modulo all divisors of some modulus $n$, i.e. of $\text{ord}_d(a)$ with $d\mid n$? It is easy to see that it is a ...
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### Reviewing a proof to show that a subgroup of a cyclic group is also cyclic [closed]

I want to prove the following statement: Let $H$ be a subgroup of a cyclic group $G$, here, finite. Then $H$ is also cyclic. My question is about an attempt that I include below; in particular, ...
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### Is there any External Direct Product which is cyclic but it is the product of two non-cyclic groups? [closed]

Is there any External Direct Product which is cyclic but it is the product of two non-cyclic groups? I know that for an EDP, say $G\times G'$ to be cyclic the $\gcd(o(G),o(G'))=1$. But I am unable to ...
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### show G has a faithful representation of degree 1 over $\mathbb{C}$ iff it is cyclic

Show that a finite group G has a faithful representation of degree 1 over $\mathbb{C}$ iff G is cyclic. I saw the following related post, but I'm not sure if I fully understand it: Faithful ...
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### What is a group and its operation?

I am learning group theory by myself and has not reached cyclic groups yet but as I have read cyclic groups are group generated by a single element and is denoted as $G = \langle x, *\rangle$ where $*$...
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### Prove that when $|H| = \infty$, $\langle x^a \rangle \neq \langle x^b \rangle$ and $\langle x^m \rangle = \langle x^{|m|} \rangle$

I want to prove Theorem 7(2) in section 2.3 from Abstract Algebra 3rd Edition by Dummit & Foote. This theorem states: Let $H = \langle x \rangle$ be a cyclic group. If $|H| = \infty$, then for ...
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### Invariants of the Hyperoctahedral group

Apologies in advance for what I am asking might be too trivial, I am not a mathematician. I have a function $V(x_1,\dots,x_n)$ that could have, at most, a hyperoctahedral (signed permutations) ...
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### Invariant factors and elementary divisors of quotient group

I need help solving the following problem: Let $G = \Bbb{Z}_9 \times \Bbb{Z}_9 \times \Bbb{Z}_9$ be the product of three cyclic groups of order 9. Give invariant factors and elementary divisors of ...
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### Why is pointwise stabilizer of a reflecting hyperplane a cyclic group?

Let $G$ be a pseudoreflection group and $H$ be a reflecting hyperplane. Let $G_H$ be the subgroup of $G$ consisting of all those elements of $G$ which stabilize the reflecting hyperplane $H$ pointwise....
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### What is the isomorphism class of the kernel of a homomorphism of cyclic groups?

Consider the homomorphism $\phi:(\mathbb{Z}/27\mathbb{Z})\times(\mathbb{Z}/9\mathbb{Z})\to\mathbb{Z}/3\mathbb{Z}$ defined by $(a,b)\mapsto a+b$. What is the isomorphism class of $\ker\phi$? The ...
1 vote
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### Isomorphism of cyclic groups $U(p)$ and $\mathbb{Z}_{p-1}$ [closed]

For prime $p$ the group $U(p)$ is isomorphic to $\mathbb{Z}_{p-1}$. Considering the case for $U(19)$ it is isomorphic to $\mathbb{Z}_{18}$. Since isomorphism preserves order of elements so order of an ...
1 vote
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### What should I type in Archive.org search bar to find books covering proof of $U_{p^α}$ is cyclic (where $p$ is an odd prime)?

Let $p$ be an odd prime number. The multiplicative group modulo $p^α$, $U_{p^α}$, is cyclic for all $α\geq 1$. I have read proofs of this result in Vinogradov "Elements of number theory" (...
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### Every automorphism of $Z_n$ is equal to some $\sigma_a$

Dummit and Foote 2.3 Ex. 26 I was working on this problem, and I'm not convinced of part c. Since $Z_n$ is the cyclic group of order $n$, $Z_n = \langle x \rangle,$ so every element of $Z_n$ is of the ...
1 vote
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### Vinogradov's proof that $U_{p^{\alpha}}$ is cyclic. How to prove $p^{r-1} (p-1) \mid \delta$ for $1 \leq r \leq \alpha$

The proof of Vinogradov that $U_{p^{\alpha}}$ ($\alpha \geq 1$) is cyclic for an odd prime $p$, has a part that I don't understand. We take $g$ a primitive root of $U_p$. And, at a certain point of ...
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### Non-constructiveness and finite mathematics

It is known that the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^{\times}$ is cyclic. However, known proofs are all "non-constructive" in the sense that they don't rely on a direct ...
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### A cyclic group of order $n$ has a unique subgroup of order $d$ for each $d\mid n$

On p.25 of the revised third edition of Serge Lang´s Algebra, he proves that any cyclic group $G$ of order $n$ has a unique subgroup of order $d$ for each $d\mid n$ and his proof is as follows: Proof. ...
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1 vote