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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18 views

Cyclic Groups - $a^k = e \text{ iff } n|k$

I saw this proof in the book on Abstract Algebra. Here is part of it: Let $G$ be a cyclic group of order $n$ and $a$ is the generator of $G$. Then $a^k = e \iff n|k$ Proof: Suppose $a^k=e$. By the ...
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1answer
72 views

How to prove that G is a cyclic Group? [duplicate]

Suppose $G$ is a finite Abelian group and, $\forall\ n\in \mathbb{N} $, there exist at most $n$ elements in $G$ which satisfy $x^n=1$. Prove $G$ is cyclic. Thanks for your help.
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If $G$ is a Finite Group such that $H\le K$ or $K\le H$ for all Subgroups $H,K$ of $G$, then $G$ is Cyclic and of order $p^n$ for some Prime $p$.

Since each subgroup $K$ is contained in some other subgroup $H$, we can list the subgroups of $G$ in ascending order $$\lbrace 1 \rbrace < G_1< G_2 < G_3\cdots< G_{k-1}<G$$ By ...
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1answer
40 views

About a proof regarding a property of groups of order $pq$ where $p$ and $q$ are primes

I'm studying right now Automorphisms in Dummit & Foote's Abstract Algebra (Section 4.4). In pages 135-136, the following example is given: and here's Proposition 16 muntionned in the ...
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0answers
47 views

Artin's Algebra, Exercise 2.4.11. (1st edition)

I have been working through Artin's algebra book. Here's a simple exercise, but I want to make sure I am not missing anything important. Let $(G,\cdot)$ be a group and let $x,y \in G$ with orders ...
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1answer
14 views

The number of cyclic subgroups

Let $|G|=p^{n} $, then for every $d$, that $d|p^{n} $, there are cyclic subgroups of order $d$ for group $G$. These subgroups be as $G_{0} \subseteq G_{1} \subseteq ...\subseteq G_{n} =G$,where ...
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1answer
32 views

Without using Cauchy's or Sylow theorems ; can we prove that every group of order $65$ is cyclic? [duplicate]

Without using Cauchy's or Sylow theorems, can we prove that every group of order $65$ is cyclic? Please help, thanks in advance (any technique of group homomorphisms and normal subgroups can be used). ...
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29 views

Let G be an abelian group. Let V be an irreducible faithful CG-module. Prove that dimV = 1 and G is cyclic.

I was wondering if I could get some help with the following problem. I know how to prove it with Schur's Lemma but I'm having problems without it. Let G be an abelian group. Let V be an irreducible ...
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2answers
39 views

Prove that Euler phi function is multiplicative by a given theorem

I had proven a theorem which states that If $G=\langle a\rangle$ has order $rs$ , where $(r,s)=1$. Then there are unique $b,c\in G$ with $b$ of order $r$, $c$ of order $s$ and $a=bc$. There is ...
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1answer
62 views

Let $G$ be a group and $H\le G$ and $a\in G$. which ones are always correct?

Let $G$ be a group, $H\neq G$ and $H\le G$ and $a\in G$. which ones are always correct? a) $|a|=|a^{-1}|$ b) H and G has the same unit c) $|H|\neq0$ d) if $a^{12}=a^2$ then |a| is even ...
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2answers
36 views

Cyclic subgroups and their cyclic groups [closed]

If a subgroup $\langle a\rangle$ is cyclic, then is the group that contains this cyclic subgroup also cyclic?
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1answer
29 views

Show that $U(\mathbb Z_{p^n})$ is cyclic by considering the order of $1+p$

Show that $U(\mathbb Z_{p^n})$ (the group of units) is cyclic for $p$ an odd prime, $n \in \mathbb N$. We are given a hint to consider the order of $1+p$ in this group. I have no idea how this leads ...
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1answer
35 views

Trying to prove that a group is Cyclic

Suppose that the order of $G$ is divisible by at least two distinct primes. Also, let $g\in G$ that order of $g$ is divisible by every prime divisor of $o(G)$ and $\forall x\in G$, $o(x)\mid o(g)$ ...
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34 views

How do I interpret “K2 mod K1”?

I've been doing a bit of work these past couple of nights on computing cyclic subspace decompositions, finding cyclic bases, and then computing the Jordan canonical form of matrices. My question is: ...
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0answers
27 views

Subgroups of the multiplicative group of a Field [duplicate]

I am reading S. Roman's book "Field Theory". In chapter $1$ I found the following exercise Let $F^*$ be the multiplicative group of all nonzero elements of a field $F$. We have seen that if G is a ...
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1answer
28 views

Why does the Principle of Well-Ordering imply a remainder of $0$ for the division algorithm?

I'm currently reading a text (Thomas W. Judson, Abstract Algebra - Theory and Applications) where the author proofs the theorem that every subgroup of a cyclic group is cyclic. The proof goes as ...
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1answer
17 views

Does there exist any non-trivial square matrices of dimension $n$ with power cycles of less than $n$

Earlier I was faced with the matrix: $$A=\begin{bmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix}$$ Which cycles ...
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0answers
31 views

Which abelian groups have only a single composition series?

Cyclic groups of composite powers don't: for example, $1=C_1\triangleleft C_3\triangleleft C_6 $ and $1=C_1\triangleleft C_2\triangleleft C_6 $ are both composition series for $C_6$. But cyclic ...
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2answers
44 views

Finding subgroups of $\mathbb{Z}_{13}^*$

I need to find all nontrivial subgroups of $G:=\mathbb{Z}_{13}^*$ (with multiplication without zero) My attempt: $G$ is cyclic so the order of subgroup of $G$ must be $2,3,4,6$ Now to look for ...
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1answer
27 views

Group theory exercise from “Rational Points on Elliptic Curves”

Let $A$ be an abelian group and for every $m \geq 1$, let $A_m$ be the set of elements $P$ of $A$ such that $mP=0$. Now, suppose that $A$ has order $M^2$ and that for every integer $m$ dividing ...
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2answers
67 views

Subgroups of $\mathbb{Z}_{13}^*$ using “GAP language”

How can I know the nontrivial subgroups of $\mathbb{Z}_{13}^*$ (with multiplication without zero) using GAP language
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2answers
45 views

Show that $G:=\mathbb{Z}_{13}^*$ is cyclic

I need to prove that $G:=\mathbb{Z}_{13}^*$ (without zero with multipcation)is cyclic My attempt: I tried to check each element in $G$ if it is a generator or not: $$ \begin{align} &1^1=1\mod ...
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26 views

Verify that $A \oplus B$, where $A$ and $B$ are cyclic groups of orders 2 and 3, is the cyclic group of order 6

Let's define $A$ and $B$ as follows: $A$ = {e,a} $B$ = {e,b,2b} Then $A\oplus B= \{\{e+e\},\{e+b\},\{e+2b\},\{a+e\},\{a+b\},\{a+2b\}\}$ which is equal to ...
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1answer
59 views

Every group of order $35$ is cyclic? [duplicate]

Prove that every group of order $35$ is cyclic. Now, the subgroups of this are ones whose orders divide the order of this group(by lagrange), these are of prime orders $7$ and $5$. and I guess ...
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3answers
47 views

If $H$ and $K$ are subgroups of $G$ whose orders are coprime. Is $H\cap K$ a subgroup of $H$ and $K$?

If $H$ and $K$ are subgroups of $G$ whose orders are coprime. Is $H\cap K$ a subgroup of $H$ and $K$? They both have the same identity, so we know at minimum we have $\{e\}$ so it is the trivial ...
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1answer
30 views

The generators of the group $\langle\mathbb{Z}_n,\oplus\rangle$ are all $g \in \mathbb{Z}_n $ for which $\gcd(g,n)=1$ [duplicate]

I'm trying to find a proof of this: The group $\langle\mathbb{Z}_n,\oplus\rangle$ is cyclic for every $n$, where $1$ is a generator. The generators of the group ...
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2answers
83 views

Proof to a property of Euler's totient function

The property is $$\sum_{d|n}\phi(d) = n$$ And the proof provided is If $d$ divides $n$, let $C_d$ be the unique subgroup of $\mathbb{Z}/n\mathbb{Z}$ of order $d$, and let $\Phi_d$ be the set of ...
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83 views

Does group $G$ of order 42 have a normal cyclic subgroup of order 21?

Show that a group $G$ of order 42 has a normal cyclic subgroup of order 21. What I did so far is using Sylow's theorem to show that $G$ has a unique 7-sylow subgroup $S(7)$ (which is normal) and {1 ...
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13 views

How do you find the order of a cyclic group?

What is the order of the cyclic group generated by $(1 2 5)(3 4)$? What is the order of the cyclic group generated by $(1 2 5)(3 5)$? I've looked through my notes and can't find notes on this and can ...
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1answer
41 views

Order of a cyclic group?

When finding the order of a cyclic group, do we determine so by counting the number of elements in that group generator by the cyclic group?
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98 views

In $(\mathbf Z/p^r \mathbf Z)^*$, finding an element with order $p-1$.

Let $p$ be an odd prime number. I want to prove that $(\mathbf Z/p^r \mathbf Z)^*$ is an cyclic group. I have known that $\overline {p-1} \in (\mathbf Z/p^r \mathbf Z)^*$ is of order $p^{r-1}$. Since ...
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If $G = \langle a\rangle$ and $b$ $\in$ $G$, the order of $b$ is a factor of the order of $a$

Prof. Pinter's "A Book of Abstract Algebra" presents the following exercise from the "Cyclic Groups" chapter: If $G = \langle a\rangle$ is finite and $b$ $\in$ $G$, the order of $b$ is a factor ...
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List of the Elements of $(1 \rightarrow 6,2 \rightarrow 1, 3 \rightarrow 3, 4 \rightarrow 2, 5 \rightarrow 5, 6 \rightarrow 4)$

Dr. Pinter's A Book of Abstract Algebra presents the following exercise in the "Cyclic Groups" chapter. List the elements of $\langle f\rangle$ in $S_6$ where $f$ = $$(1 \rightarrow 6,2 ...
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Proof: Every Cyclic Group is Abelian

Dr. Pinter's "A Book of Abstract Algebra"'s chapter on Cyclic Groups presents the exercise: Prove that every cyclic group is abelian. Here's my attempt: By Theorem 1 (of this chapter): ...
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1answer
37 views

Correctness of proof of generalized Euler's criterion

My lecture notes on quadratic residues were pretty sloppy, and I was trying to prove some theorems from class. I don't think I'm correct, though. Can anyone tell me if I'm wrong? Specifically, I think ...
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Abstract Algebra: Every group has a cyclic subgroup

I have to show that every group has a cyclic subgroup. I know what this means, and to me it is obvious, yet I am not sure how to formally write it. I proved it directly, as follows: Let G be a ...
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1answer
46 views

under following conditions $G$ has only one subgroup of order $p$.

Let $|G|=p^m$ for $m \ge 2$. If every subgroup of $G$ of order $p^2$ is cyclic, then $G$ has only one subgroup of order $p$.
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139 views

A finite group which has a unique subgroup of order $d$ for each $d\mid n$.

Problem Suppose G is a finite group of order $n$ which has a unique subgroup of order $d$ for each $d\mid n$. Prove that $G$ must be a cyclic group. My idea: I try to prove it by induction. Let ...
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1answer
40 views

Is there any cyclic subgroup of order 6 in in $ S_6$?

Is there any cyclic subgroup of order 6 in $ S_6$? Attempt: $|S_6|=6!=720$ Let $H$ be a subgroup of $S_6$ ,$H$ cyclic $\iff\langle H \rangle=\{e,h,h^2,...,h^{n-1}\}=S_6$
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Abelian group which is not one of these

Im struggling to find a finite abelian (commutative , associative) group $(G,\circ)$ with some specific conditions: $a\circ b$ isn't naive addition $a+b$ for $a,b\in G$ $G$ is a subset of ...
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1answer
30 views

Cyclic consecutive zeros of binary sequence with prime length

I found a feature that if $N>5$ is a prime, and $M \triangleq \frac{N-1}{2}$ is also a prime, then we will always have a binary sequence $x_1,\ldots, x_N$ with $L=\frac{N-1}{2}$(or ...
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If $G$ is cyclic then $G/H$ is cyclic?

If $G$ is cyclic, then $G/H$ is cyclic? The proof I got goes like this: $G$ is cyclic, so $G=<g>$ for some $g\in G$. So any coset in $G/H$ would be of the form $Hg'=Hg^n$ for some $n$. So ...
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1answer
24 views

Find the Existence and Uniqueness of a Cyclic subgroup

Giving that $|H|$ is cyclic. If $|H|=n$ then for each $a>0$ such that $a|n$ there exist a unique subgroup of $H$ of order $a$. This subgroup is Cyclic subgroup $\langle x^d\rangle$ where $d= ...
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45 views

Cyclic groups and other groups

Ok So I know that a cyclic group is a group that is generated by a single element like $\large{(Z_n,+)}$. Now I was wondering that if every group has a generator , and I found that the answer is yes ...
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Writing a group as a product of cyclic groups

How do I go about expressing a group as a product of cyclic groups? For example, express: $$O^*_K = \{\pm (24 + 5\sqrt{23})^r : r\in \mathbb{Z} \}$$ as a product of cyclic groups ($O^*_K$ is the ...
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3answers
57 views

Isomorphism of a product $C_n \times C_m$ of cyclic groups with the cyclic group $C_{mn}$

Given that $C_n$ is a cyclic group of order $n$, what conditions must integers $n$ and $m$ satisfy such that the group $C_n \times C_m$ is isomorphic to C$_{mn}$? So I attempted to investigate a ...
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1answer
64 views

$g$ has order $n$, then $\langle g\rangle=\langle g^2\rangle=\cdots=\langle g^{n-1}\rangle$

If $g$ has order $n$, then $\langle g\rangle=\langle g^2\rangle=\cdots=\langle g^{n-1}\rangle$. This should be fairly easy but somehow I just couldn't prove it. I only managed to prove the case ...
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1answer
32 views

Groups and Subgroups elements

Let $G$ be a cyclic group order $12$ with $G=\left<a\right>$. Let $H=\left<a^3\right>$. List the elements of $H$ and find the cosets. I am lost as to what the elements of $H$ would be. ...
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2answers
67 views

$|G|=p_1p_2p_3$ distinct primes with $p_i \nmid p_j-1$ then $G$ is cyclic

Problem Let $p_1,p_2,p_3$ be three distinct primes with $p_i \nmid p_j-1$ for all $1\leq i,j \leq 3$ and let $G$ be a group of order $p_1p_2p_3$. Show that $G$ is cyclic. I've tried to come up with ...
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1answer
49 views

Is the direct product $\Bbb Z \times \Bbb Z$ with operation $(n,m)+(p,q):=(n+p,m+q)$ a cyclic group?

Is the direct product $\Bbb Z \times \Bbb Z$ with operation $(n,m)+(p,q):=(n+p,m+q)$ a cyclic group? I know its not a cyclic group but how would i show this in a formal way?