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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About semi-direct product of two cyclic groups

The following question is related to seeing semi-direct products as subgroups: Let $G$ be a non-nilpotent group and $U = \langle x \rangle$ be a cyclic characteristic subgroup of the Fitting subgroup $...
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Lifting map from finite cyclic group to integers

Suppose $\phi:G \to \mathbb Z/n\mathbb Z$ is a group quotient with finite cyclic image. Under what conditions can $\phi$ be lifted to a homomorphism $\tilde \phi:G \to \mathbb Z$? Specifically I'm ...
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What does the book mean? Confusion over element $a$ in theorem $\langle a^k\rangle=\langle a^{(n,k)}\rangle$

This question may seem silly, but it is important for me. My background: Non-native english speaker and first year college student working abstract algebra by myself. Theorem: Let $a$ be an element ...
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But what is with the other cyclic groups? Doesn't one also have to consider them? [closed]

I'm currently reading a textbook about abstract algebra. There is a proof that every subgroup of a cyclic group is cyclic. This proof is using the fact as every proof I have found on the Internet ...
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generalized version of definition of cyclic groups and its abelian property [closed]

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= \...
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Schur's Multiplier exercise (Problem 5A.7 Isaacs' Finite Group Theory)

I have a question about the following problem [Finite Group Theory, Martin Isaacs, Chapter 5]: Let $ B $ and $ C $ be cyclic subgroups of a finite group G, and suppose that $ BC = G $ and $ B \cap C &...
Elianna 's user avatar
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$p$-group with a cyclic subgroup

Throughout studying a paper about finite $p$-groups, I have the following question Let $G$ be a finite $p$-group with nilpotency class 3 and $\gamma_i(G)$ denote the i'th term of the lower central ...
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extension condition for free abelian groups

if $G$ is a free abelian group with basis {${a_\alpha}$} then given the elements {${y_\alpha}$} of an abelian group $H$, there are homomorphisms $h_\alpha : G_\alpha \to H$ such that $h(a_\alpha)=y_\...
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Non-cyclic subgroup of order 4 in non-dihedral group

A group $G$ has sixteen elements: $$\{e, r, r^2, \dots , r^7, s, rs, r^2s, \dots , r^7s\},$$ where $r$ and $s$ satisfy the relations $r^8 = e, s^2 = e, sr = r^3s$. (Note that $G$ is not a dihedral ...
mathussy's user avatar
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representation of an element in a cyclic group in a normal form

Suppose $A$ is a finite cyclic group of order $n$, and $a$ is an element in $A$ of order $d$. My question is: Question 1:Can we find a generator $x$ of $A$ such that the element $a$ can be written in ...
Xiaobo Zhuang's user avatar
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An abelian group of order 35 must be cyclic [duplicate]

I'm reading through Contemporary Abstract Algebra by Joseph A. Galian and I stumbled upon the following problem: Suppose $G$ is an Abelian group of order $35$ and every element of $G$ satisfies $x^{...
Selim Bamri's user avatar
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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 ...
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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 ...
The Wanderer's user avatar
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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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Group of order 1001 is cyclic. (Basic Group Theory Solution) [duplicate]

I am trying to prove that if $G$ is a group of order $1001$, then $G$ has normal subgroups of order $7$, $11$, and $13$, and thus $G$ is cyclic. I know that by Cauchy's Theorem, there exist elements $...
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How to find subgroups of direct product of cyclic groups [duplicate]

Find all subgroups of $\mathbb{Z_2} \times \mathbb{Z_5} \times \mathbb{Z_7}$. I know that this would be isomorphic to $\mathbb{Z_{70}}$, so that I would be looking for $8$ subgroups because $70$ has $...
MrMustache's user avatar
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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 ...
Ayush Kumar Singh's user avatar
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1 answer
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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 ...
utx7563yu's user avatar
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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 ...
user2867280's user avatar
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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 ...
Jonathan McDonald's user avatar
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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 ...
niobium's user avatar
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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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Confusion about the proof of $\langle a^k\rangle=\langle a^{(n,k)}\rangle$ [closed]

This theorem is from Contemporary Abstract Algebra by J. Gallian page $78$. Let $a$ be an element of order $n$ in a group and let $k$ be a positive integer. Then $\langle a^k\rangle=\langle a^{(n,k)}\...
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How can I see that $o(G) = p^n$ without using the fundamental theorem of finitely generated abelian groups?

Let $G$ be a finite group such that for all pair of subgroups $H, K \leq G$ we have $H \subseteq K$ or $K \subseteq H$. I want to prove that there exist $p \in \mathbb{N}$ prime and $n \in \mathbb{N}$ ...
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Is this a valid proof? Group Theory

I want to show that $(\mathbb{Q}^{+}, \cdot)$ is not cyclic. Suppose that $0<q<1$ a generator for $(\mathbb{Q}^{+}, \cdot)$. Then, for every $a \in \mathbb{Q}^+$ there is a $k \in \mathbb{Z}$ ...
ends7's user avatar
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Calculate product of disjoint cycles

I'm working on the following problem: Let $a = (1, 5, 3)(2, 4)$ and $b = (2, 4, 5)$ be permutations in $S_6$. Calculate the product $ba$. Answer with a product of disjoint cycles. $ba $ This is what I ...
Kei Len's user avatar
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Isomorphisms between cyclic groups

I was solving an exercise which asked to determine all abelian groups of order 48 two by two not isomorphic with each other and it seemed natural to me to use in the process the following proposition: ...
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3 answers
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Is only one generator enough to find other generators?

I am self learner high school student interested in abstract algebra,nowadays. I saw the followings in my book: "If a is a generator of a finite cyclic group G of order n, then the other ...
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$a$ and $b$ are elements of the group $G$, then if $a \in \langle b \rangle$, then $\langle a\rangle \subseteq \langle b\rangle$

Assume that $a$ and $b$ are elements of the group $G$, then if $a \in \langle b\rangle $, then $\langle a\rangle \subseteq \langle b\rangle$ How can I prove the foregoing identity? It is from the ...
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To prove that two cyclic subgroups H and K within a cyclic group G are the same when their orders are equal

I see in Fraleigh's abstract algebra book, there is a theorem stating that two cyclic subgroups $H$ and $K$ of a cyclic group $G$, denoted as <$a^s$> and <$a^t$>,respectively, are the same,...
Gao Minghao's user avatar
2 votes
0 answers
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$a$ is a torsion element of the group $G$, $yay^{-1}=a^k \neq a$. Can we conclude that $\langle a,y\rangle $ is a finite group?

Assume that $G$ is a group. Also, $a$ is a torsion element of $G$ such that $a^m=1$, for some natural number $m$. Also, there exists an element $y \in G $ such that $yay^{-1}=a^k \neq a$, when $gcd(k,...
Reza Fallah Moghaddam's user avatar
1 vote
2 answers
122 views

What is the intuition for recognizing cyclic groups?

Compare these two sets - $\{ \pi^{n}: n \in \mathbb{Z} \}$ --- Cyclic $\{ 2^n 3^m: m,n \in \mathbb{Z} \}$ --- Not cyclic. In the first case $\{ \pi^{n}: n \in \mathbb{Z} \}$ could be expanded to $\{ ...
Parth Gupta's user avatar
5 votes
4 answers
257 views

Show that a subgroup of order $3$ is normal in a group of order $15$

I've been trying to show that any group of order $15$ is cyclic and the only missing part in my proof is to show that the subgroup with $3$ elements (which we know exists by Cauchy's theorem) is ...
ephe's user avatar
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Let $G=\left\langle a \right\rangle$ Show that any subgroup of G is also cyclic [duplicate]

Suppose H is an arbitrary subgroup of G and $H\neq \{e\}$. Let $a^{k_{0}}\in H$. Let $b=a^{t}$ be an arbitrary member of H such that $k_{0}\le t$. For some $q,r$ s.t. $t=k_{0}q+r$, then $a^{t}=a^{k_{0}...
Not Friedrich gauss's user avatar
3 votes
1 answer
132 views

Let $G_1, G_2$ be non-trivial groups. If $G_1 \times G_2$ is cyclic, then $G_1 \times G_2$ is finite.

Let $G_1, G_2$ be non-trivial groups. If $G_1 \times G_2$ is cyclic, then $G_1 \times G_2$ is finite. I'm asked about the veracity of this statement and I did the following: suppose BWOC that $G_1 \...
J P's user avatar
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6 votes
1 answer
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Proof by contradiction using properties of normal subgroup: A group is cyclic if $x^m=e$ has at most $m$ solutions.

Statement: Let $G$ be a finite group. Show that if for each positive integer $m$ the number of solutions $x$ of the equation $x^n=e$ in $G$ is at most $n$, then $G$ is cyclic. There are various ...
Nothing special's user avatar
0 votes
2 answers
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Infinite non cyclic group with proper subgroups being cyclic

I was thinking about the question regarding infinite groups with all of its proper subgroups being cyclic but the group itself is not cyclic. The finite case is a well known fact $(K_4)$. I have found ...
nkh99's user avatar
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Are all the virtual cyclic subgroups of $PSL(2,\mathbb{R})$ elementary?

Let $\Gamma$ a virtual cyclic subgroup of $PSL(2,\mathbb{R})$, i.e. there are $\Lambda \leq \Gamma$ with $\Lambda$ cyclic of finite index. ¿Is $\Gamma$ elementary?, where elementary means that there ...
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