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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20 views

can composite groups have primitive roots (be cyclic)?

Imagine Z/nZ with n not prime (n=pq). Can the multiplicative group be cyclic? I read the paper over here: http://math.uga.edu/~pete/4400primitiveroots.pdf At one point he dismiss the case where the ...
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0answers
23 views

Show that ${C_\infty }/\left\langle {{c^n}} \right\rangle \simeq {C_n}$

Let ${C_\infty } = \left\langle c \right\rangle $ be an infinite cyclic group. Show that if $n > 0$, then $${C_\infty }/\left\langle {{c^n}} \right\rangle \simeq {C_n}$$where ${C_n}$ is a ...
2
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1answer
27 views

Prove that for any n in $ \Bbb N $ there exists a unique cyclic subgroup $H_n$ $\subset$ G of order n. Where G is the group $\Bbb Q/$ $\Bbb Z$.

Prove that for any n in $ \Bbb N $ there exists a unique cyclic subgroup $H_n$ $\subset$ G of order n. Where G is the group $\Bbb Q/$$\Bbb Z$. Existence was not difficult to show: Let $H_n$ = < ...
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2answers
56 views

Can Lagrange's Theorem for algebraic structure apply here?

For a positive integer $n$ let $Φ(n)$ denote the number of elements $r∈\mathbb Z_n$ such that $\gcd(r,n)=1$. Show $Φ(mn)=Φ(m)Φ(n)$ for all $m, n∈\mathbb N$ such that $\gcd(m,n)=1$. The only thing I ...
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2answers
23 views

Question to the proof of: Let $A$ be a finite abelian group and let $g \in A$. Suppose that $\chi(g)=1$ for every $\chi \in \hat A$. Then $g=1$.

Good day, Currently I am working with the book "A First Course in Harmonic Analysis" by A. Deitmar and I am stuck in the beginning of Chapter 5 on the proof to Lemma 5.1.5. I am repeating the few ...
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0answers
11 views

Conway's theorem on the number of orbits on the set of all ordered cycles in a $d$-valent graph

I am trying to understand Conway's theorem on the number of orbits on the set of all ordered cycles in a $d$-valent graph. I quote it from Cycles in graphs and groups by Kantor. Theorem $1$ ...
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1answer
21 views

Proving existence of automorphism

Let $G$ be a cyclic group, and $a,b$ two generators of $G$. Prove that exists automorphism $f$ such that $f(a)=b$ and that if $f$ is an automorphism then $f(a)$ generates $b$. $G$ is cyclic, thus ...
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0answers
17 views

Showing F-matrix representation is irreducible over $\mathbb{R}$

I have $G$ the cyclic group of order 4 and its $F$-(matrix) representation $T$ is $$\hat{T}(g) = \bigg[ \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \bigg]. $$ I am trying to show that ...
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0answers
60 views

If $G/Z(G)$ is cyclic, why is $G$ only abelian and not also cyclic?

If the factor group with respect to the center of $G$ is cyclic, then $(aZ(G))^n=gZ(G)$ for some $n$ and any $g$, where both $a$ and $g$ are from $G$ (and $a^n$ is, too). Because of the definition of ...
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1answer
38 views

Show that $|a^{k}|=|a^{n-k}|$

Let G be a group and let $a$ be an element of G of order $n$. For each Integer $k$ between $1$ and $n$, show that $\left | a^{k} \right |=\left | a^{n-k} \right |$ My attempt is as follows: $\left ...
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0answers
19 views

Element of infinite order and all generator of subgroups

Suppose that a has infinite order Find all generators of subgroup $\left \langle a^{3} \right \rangle$ Now, since a has infinite order then so does $\left ( a^{3} \right )^{n} $for if a has ...
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0answers
17 views

showing the inverse of an element is a generator of the non-inverse

Question: Let $G$ be a group and let $a \in G$. Prove that $\left \langle a^{-1} \right \rangle=\left \langle a \right \rangle$ Suppose $\left \langle a^{-1} \right \rangle$ so $\left ...
1
vote
1answer
20 views

Proof for generator of the group of integer under addition modulo

Theorem: An integer $k$ in $\mathbb{Z}_{n}$ is a generator of $\mathbb{Z}_{n}$ If and Only if $gcd\left ( n,k \right )=1$ My problem lies with proving the "If" condition and here is my ...
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0answers
25 views

Proof for generators of cyclic group [duplicate]

Theorem: Let $G=\left \langle a \right \rangle $ be a cyclic group of order n. Then $G = \left \langle a^{k} \right \rangle$ if and only if $gcd\left ( n,k \right )=1$ I've proven the "only ...
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0answers
22 views

Consider a set consisting of matrices and show it is a group.

I am giving $$A=\{1,i,-1,-i\}$$ and $$B=\{\begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}, \begin{pmatrix} 0 & -1 \\ 1 & \phantom{-}0\end{pmatrix}, \begin{pmatrix} -1 & \phantom{-}0 \\ ...
4
votes
1answer
41 views

Subgroups of the Semi-Direct Product $\mathbb{Z}/\mathbb{7Z} \rtimes (\mathbb{Z}/\mathbb{7Z})^{\times}$

I want to list all the subgroups of the semi-direct product $\mathbb{Z}/\mathbb{7Z} \rtimes (\mathbb{Z}/\mathbb{7Z})^{x}$, under the homomorphism $\theta: (\mathbb{Z}/\mathbb{7Z})^{\times} \rightarrow ...
3
votes
0answers
32 views

Proving that a Galois group is cyclic

Let $K$ be a field containing a primitive $n$th root of unity and let $F = K(t)$ be the field of rational functions over $K$. I'm having trouble proving that for each $n > 1$ the field $F$ is Galois ...
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1answer
9 views

Suppose that $H$ is a (cyclic) subgroup of order $m$ of a cyclic (abelian) group $G$ of order $n$. What is $G/H$?

Suppose that $H$ is a (cyclic) subgroup of order $m$ of a cyclic (abelian) group $G$ of order $n$. What is $G/H$? This is taken from an exercise at the end of a section that I must have read 6 ...
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0answers
18 views

Number of elements in factorgroup

Consider the following Group Theory question: Find the number of elements in each of the factorgroups: $$ \frac{\mathbb{Z}/6\mathbb{Z}}{\left \langle \bar{2} \right \rangle}, ...
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0answers
20 views

G is cyclic if it is a finite p-group and has only one maximal subgroup [duplicate]

How can I show that for a finite $p$-group $G$,$G$ is cyclic if it has just one non trivial maximal subgroup?
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1answer
23 views

Proof groups equality: $<x^{t_1},x^{t_2}> = <x^{gcd(t_1,t_2,n)}>$

Let $C_n$ be a cyclic group of order n. I need to proof $<x^{t_1},x^{t_2}> = <x^{gcd(t_1,t_2,n)}>$ where gcd is the greatest common divisor of $t_1,t_2,n$ and $<>$ denotes the ...
1
vote
1answer
48 views

$\sum_{d|n} \varphi(d)=n$

I want to solve $\sum_{d|n} \varphi(d)=n$ using Group theory. Here, $\varphi(d)$ is Euler's totient function. I think I should use $\Bbb Z_n$ and fundamental theorem of cyclic group. Then I use ...
2
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1answer
259 views

Structure of $Gal(\mathbb{Q}(\zeta_{15})/\mathbb{Q})$?

$\zeta_{15}$ is $15$th primitive $n$th root of unity. Question: Find the structure of the group $Gal(\mathbb{Q}(\zeta_{15})/\mathbb{Q})$ I know that if $p$ is prime then ...
0
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1answer
28 views

Cyclic extension of degree $2$ of $\mathbb{F_2}$

I want to find a cyclic extension of degree $2$ of $\mathbb{F_2}$ The degree of its minimal polynomial is $2$, i.e. a quadratic polynomial. The only cyclic field of degree $2$ which I can think of ...
1
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1answer
28 views

List all subgroups of $\mathbb Z_6$ and $\mathbb Z_8$

List all the subgroups of $\mathbb Z_6$ and $\mathbb Z_8$. I think this implies that the operation is addition because that makes the sets above groups. I was thinking that for $\mathbb Z_6$, the ...
0
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1answer
33 views

Sum of two cyclic codes is a cyclic code.

My text made a small comment about cyclic codes: The sum of two cyclic codes $C_1$ and $C_2$, defined by: $C_1+C_2=\{c_1+c_2: c_1\in C_1, c_2\in C_2 \}$ The sum of two cyclic codes is also a cyclic ...
2
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1answer
58 views

How to prove for a finite group that $| \langle g \rangle|=o(g)$? [duplicate]

How to prove for a finite group that $|\langle g \rangle|=o(g)$? I really don't know how to show that this is true I tried to say let $o(g)=m$ and then show that $|\langle g \rangle|$ has exactly $m$ ...
0
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0answers
32 views

What is the generator of the group of the units in the field $\mathbb{F}_3[x]/(x^3+2x+2)$?

I have shown that $\mathbb{F}_3[x]/(x^3+2x+2)$ is a field. Then the question asks what is the generator of the group of the units? no idea how to proceed from there....
2
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1answer
29 views

Is there a strategy for expressing finitely generated abelian group as the direct sum of cyclic groups?

I know that every finitely generated abelian group can be expressed as a direct sum of cyclic groups. I wondering how easily we can find the cyclic groups given an abelian group. Specifically, one of ...
0
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2answers
35 views

Proving a subgroup is cyclic

Let $G=Z_n$ for $n\gt1$ and let $a,b \in G$ where $a,b$ are two integers (with at least one nonzero). Prove that the subgroup of G generated by $a$ and $b$ is indeed cyclic and is generated by $c \in ...
4
votes
1answer
72 views

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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0answers
19 views

Struggling with $S_p$ can be generated by $(12)$ and $(12 \dots p)$?

We can decompose any element of $S_p$ into the form $(a_1 \ \ b_1)(a_2 \ \ b_2) \dots (a_i \ \ b_i)$. If for some $1 \le j \le i$, $|a_j - b_j|=1$, it's easy! Because: Let $\sigma = (12 \ldots p)$. ...
0
votes
1answer
32 views

Prove a certain cyclic extension with prime power order is simple

Let $E/F$ be a cyclic extension of degree $p^n$, where p is prime. Let $L$ be an intermediate field such that $[E:L] = p$. If $E = L(\alpha)$, prove that $E = F(\alpha)$. I've tried to work it ...
0
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1answer
53 views

All groups where underlying set has at most 4 elements

I have this homework problem. It states: By constructing the possible Cayley tables, list all groups where the underlying set has at most 4 elements, up to isomorphisms. That is, no two groups in ...
1
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1answer
23 views

Prove for two groups, one less than the other, the smaller is a cyclic subgroup of larger.

Suppose that $H$ and $G$ are groups and that $H \le G$. Prove that if $H \cong \mathbb Z$ or $H \cong \mathbb Z_n$, then $H=\langle g \rangle$ for some $g \in G$ I'm not entirely sure where to go ...
0
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2answers
22 views

Finite cyclic group of residue set of integers

Recall that $Z_n$ is the finite cyclic group with n elements and group multiplication given by $x * y :=(x+y) \pmod n$. For $n = 5$, find all $x \in Z_n$ such that $ \langle x \rangle=Z_n$ So for ...
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3answers
44 views

Normal subgroup test

Hi there I have this problem: Is $ <p^6\epsilon^5> $ a normal subgroup of the Dihedral group $ D_4 = \{ I,p,p^2,p^3,\epsilon, p\epsilon, p^2\epsilon,p^3\epsilon \} $? Since I'm not that good at ...
0
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1answer
32 views

Determining whether a group is cyclic from its Cayley Table

How can we check if a group is cyclic by using its Cayley table? Further, how we find out the generators from the Cayley table?
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0answers
34 views

combinatorics on cyclic sequences

Given $m\geq 1$, let $I=(a_1,\ldots,a_{3m})$ be a sequence such that $I$ contains exactly $m$ zeros, $m$ ones, and $m$ twos. Given $i=1,2$ and $j\leq 3m,k\leq m$ we can define ...
0
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2answers
34 views

G is a cyclic group of cardinality of a power of a prime number [closed]

Let $G$ be a finite group for which for every subgroups $H,K$ of it we have $H\subseteq K$ or $K\subseteq H$. Prove that $G$ is a cyclic group and its cardinal is a power of a prime number.
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2answers
26 views

Why are cyclic groups of order $p$ (prime) unique up to isomorphism?

It says here that "By the classification of cyclic groups, there is only one group of each order[prime] (up to isomorphism)". What is the intuition behind this and how would you go about proving ...
8
votes
1answer
104 views

Sufficient Conditions for $(R,+,.)$ to be a commutative ring

Today, in class, my algebra professor stated this particular general result. Theorem. Let $R$ be a ring of order $pq$ where $p,q$ are two primes with $p\gt q$ and $q\not\mid (p-1)$. Then, $R$ is a ...
0
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3answers
47 views

Group of order $p^2$. [closed]

$G$ is a group of order $p^2$, $p$ is a prime. If $Z(G)=p$ then $G/Z(G)$ has order $p$ and $G$ is cyclic. Why $G$ is cyclic? Is it related to the Lagrange Theorem? Actually, I have no idea about it. ...
2
votes
1answer
152 views

Number of cyclic subgroups of the alternating group $A_8$

Find the number of cyclic subgroups groups of the alternating group $A_8$. I don't know how to even begin approaching this question. Is there a faster way to do this besides explicitly list each and ...
3
votes
2answers
100 views

Abstract: If $G$ is a group of order 10 with only one element of order 2, then $G$ must be cyclic. [closed]

Abstract Algebra: If $G$ is a group of order 10 with only one element of order 2, then $G$ must be cyclic. Thank you.
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1answer
18 views

elements in the product of subgroups in $S_4$

Let $N:=\{e,(1\,2)(3\,4),(1\,3)(2\,4),(1\,4)(2\,3)\}$ be the normal subgroup of $S_4$ and $H:=\langle(1\,2\,3\,4)\rangle$ be the cyclic subgroup of $S_4$ generated by $(1\,2\,3\,4)$. Using the Second ...
2
votes
0answers
24 views

Cyclic Subgroups of U(n) [duplicate]

from http://abstract.ups.edu/aata/exercises-cyclic.html (problem 13): "For n≤20, which groups U(n) are cyclic? Make a conjecture as to what is true in general. Can you prove your conjecture?" where ...
0
votes
1answer
18 views

For which $n$ is the group $\mathbb Z_n$ isomorphic to the subgroup of this group generated by $4$.

The exercise is as follows: Consider the group $\mathbb Z_{24}$. (a) For which $n$ is the group $\mathbb Z_n$ isomorphic to the subgroup of this group generated by the element $4$? I'm a ...
0
votes
0answers
31 views

representations of 1 dimensional complex cyclic group

Describe all the one-dimensional complex representations of the cyclic group $C_n$. Which ones are inequivalent? solution: Let $C_n$ be a cyclic group, $C_n = < g: g^n = 1>$. Then if $\phi : ...