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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Find the subgroups of A4

I had a question I was hoping for some help on: Find all of the subgroups of $A_4$ Here is what I know: $A_4$ is the alternating group on 4 letters. That is it is the set of all even ...
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3answers
37 views

Prove that $\langle a^n \rangle \bigcap \langle a^k \rangle = \langle a^{lcm (n,k)} \rangle$

Let $G$ be a group. Let $a$ be an element. Let $n,k$ be pozitive integers. Let $m$ be least common multiple of $n$ and $k$. Prove $\langle a^n \rangle \bigcap \langle a^k \rangle = \langle a^{m} ...
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1answer
9 views

Help with finding cosets for cyclic subgroups

The question I'm working on is: Let $G=\mathbb{Z}_4\times\mathbb{Z}_3\times\mathbb{Z}_2$ and consider the subgroup $H=\langle\left(0,1,1\right)\rangle$ of G. Find all cosets of H. So I know that in ...
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1answer
21 views

If $G$ has only 2 non-trivial proper subgroups H, N, then H, N are cyclic subgroup of $G$.

If $G$ has only 2 non-trivial proper subgroups H, N , then H, N are cyclic subgroup of $G$. I searched essentially same problem at If $G$ has only 2 proper, non-trivial subgroups then $G$ ...
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1answer
22 views

List the elements of a subgroup

The question is to list the elements of $H \subset \mathbb{Z}_{1610}$ if $H = \langle 1035\rangle$. I know that there are 14 elements in the set, I am not sure how to find the elements. Is the ...
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2answers
38 views

Is this group cyclic?

Is this group cyclic, and what would be its generator? where H = {a+b(sqrt(2)) element of R | a, b element of Z} I know that in order for a group to be cyclic if the generator equals the group, but ...
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2answers
119 views

Group theory question for cyclic group

I met some problem during googling. The problem and its solution are next. and I'm wondering about 2nd YELLOW BOX $$ $$ $$ $$ Why $G$ has a unique element of order 2 in case of $H=G$ ? $$ $$ ...
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1answer
73 views

Describing all the homormorphisms from $\Bbb Z_{10}$ to $\Bbb Z_{15}$

I'm working on a problem that asks me to show all the homomorphisms from $\Bbb Z_{10}$ to $\Bbb Z_{15}$. So far, my attempt is as follows: Since $\Bbb Z_{10}$ and $\Bbb Z_{15}$ are both cyclic, I ...
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1answer
31 views

Abstract Algebra: Cyclic Groups (Lattice Diagram)

Example 4.2: Lets find all the subgroups of the given group and draw the lattice diagram for the subgroup. Z12 Z36 Z8 In the book finding the subgroups is explained well but it does not explain ...
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groups and symmetry

Which of the following may be false? A) Any non-trivial element of $C_n$ generates $C_n$. B) Any subgroup of $C_n$ is cyclic. C) if $m|n$ then $C_n$ has at least one subgroup of order $m$. D) if ...
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1answer
36 views

Cyclic group generators

My question is: Can you find a cyclic group with n generators? I know that zero (or any other identity element for that matter) is included, so there would be for $Z_n$ at most n-1 generators. ...
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1answer
76 views

$G$ infinite abelian group with $[G:H]$ finite for every non trivial subgroup $H$ , to prove $G$ is cyclic

Let $G$ be an infinite abelian group such that for any non-trivial subgroup $H$ of $G$ , $[G:H ]$ is finite ; then how to prove that $G$ is cyclic ? Please don't use any structure theorem of abelian ...
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40 views

Subgroups of $S_4$ generated by cycles

I am new with abstract algebra and I trying to find all the subgroups of $S_{4}$ generated by the cycles : a) $(13)$ and $(1234)$ b) all cycles of length $3$ I am not sure how to start so I would ...
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1answer
51 views

Every subgroup is a union of cyclic subgroups of its group.

True or false? Every subgroup H of a group G is a union of cyclic subgroups of G. I think it is false,but cant think of counter example
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2answers
37 views

Prove that if $|a|=m $ and $|b|=n$ with gcd(m, n)=1 then $\langle a \rangle \cap \langle b \rangle = \{e\}$

$G$ is a group and $a, b, \in G$. So summarizing the question, if the order of $a$ and $b$ are relatively prime, then the cyclic group generated by $a$ and $b$ will only have the identity element in ...
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1answer
61 views

$\mathbb{Z}/\mathbb{2Z} \bigoplus \mathbb{Z}/\mathbb{2Z}$ not isomorphic to $\mathbb{Z}/\mathbb{4Z}$

Greetings fellow Mathematics Community. I am having some doubts about my solution to the following problem: Show that $\mathbb{Z}/ \mathbb{2Z} \oplus \mathbb{Z}/ \mathbb{2Z}$ is not isomorphic to ...
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1answer
64 views

Finite cyclic $\Bbb Z$-module and exact sequence

Suppose $M$ is $\Bbb Z$-module, cyclic, finite. How to prove $\require{AMScd}$ \begin{CD} 0 @>>> \Bbb Z @>>> \Bbb Z @>>> M @>>>0\\ \end{CD} is ...
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1answer
29 views

In S4, find all the even permutation and show that the set of odd permutations isn't stable for binary operations in S4.

I want to find the even permutations of $S_4$ so i am supposed to find the transpositions right? but of what permutation exactly do i find the transpositions? And how do i know which ones are even? ...
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3answers
29 views

GCD's and how they generate groups

I was reading something today an it was talking about $U_{15}$, all the integers relatively prime to 15, and how it was generated by the set {7,11}. I understood it all, but I thought that if the ...
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45 views

Concerning Cyclic Groups

I am new to group theory. I have a problem but I don't really understand what it is about, so I am asking somebody to explain what is the problem (I am not really seeking for solution). Here it is: ...
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18 views

The group $U_{34}$. Finding its subgroups. [duplicate]

I know U34=[1,3,5,7,9,11,13,15,19,21,23,25,27,29,31,33]. What are the proper subgroups? I know that there should be 4 of them: {1}, {$U_{34}$}. I just need to find the other 2, which will be of order ...
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4answers
52 views

How to prove that $\mathbb Z^n$ is not cyclic for $n > 1?$

The operation here is taken to be addition. Clearly $\mathbb Z$ is cyclic since $\mathbb Z = \langle 1 \rangle = \langle -1 \rangle.$ I was then looking at a question that asked if $\mathbb Z^4 \times ...
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98 views

Is $\mathbb{Z}_7^*$ cyclic?

Determine whether the following sentence is correct or not. $$ \mathbb{Z}_7^* \text{ is cyclic. }$$ Is $\mathbb{Z}_7^*$ the same as $\mathbb{Z}$ without $0$?? If it is ...
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53 views

Kernel of homomorphism between two cyclic groups of diferent order

In Malik's abstract algebra one can find the following exercise (and I paraphrase): Let $f$ be a homomorphism from a cyclic group of order 8 onto a cyclic group of order 4. Determine $\ker f$. ...
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1answer
54 views

$p$ prime, Group of order $p^n$ is cyclic iff it is an abelian group having a unique subgroup of order $p$

I 've just read from An introduction to the theory of groups from Rotman's the following theorem: Theorem 2.19. Let $p$ be a prime. A group $G$ of order $p^n$ is cyclic if and only if it is an ...
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43 views

$n$ positive integer, then $n=\sum_{d|n} \phi(n)$ (proof Rotman's textbook)

I've just read in Rotman's group theory textbook the proof of the following statement: Statement If $n$ is a positive integer, then $$n=\sum_{d|n} \phi(n),$$ where the sum is over all divisors $d$ ...
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4answers
153 views

Show that an abelian group $G$ of order 55 must be cyclic.

I know that in order to be cyclic: A group G is called cyclic if there exists an element g in G such that G = ⟨g⟩ = { $g^n$ | n is an integer } by wikipedia. But I just get lost in how simple it looks ...
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50 views

Cyclic group 60

In a cyclic group of order 60 find the elements of order 12. then find the number of element that satisfy $x^{12}=e$ So if $x^3=e$ then $x^{12}=e$ And I know $x=e$. what next do I do? Finally find ...
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40 views

A cyclic group $U_{34}$

For the elements of $U_{34}$ I know there are four subgroups. $U_{34}$ being the group generated where (a,m)=1 and m is 34. So $U_{34}$=$ [1,3,5,7,9,11,13,15,19,21,23,25,27,29,31,33]$. What are the ...
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1answer
30 views

Equality of subgroups of finite cyclic groups

My abstract algebra text has the following proof of the statement, "Let $G$ be a cyclic group with $n$ elements and generated by $a.$ Then $\langle a^s \rangle = \langle a^t \rangle \iff \gcd(s,n) = ...
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58 views

Cyclic group of order 8

In a cyclic group of order 8 show that element has a cube root. So for some $a\in G$ there is an element $x \in G$ with $x^3=a.$ Also show in general that if $g=<a>$ is a cyclic group of order ...
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58 views

Show that every proper subgroup of this group is finite.

Let $G$ be the group of rational numbers in $[0,1)$ whose denominator is a power of $2$: \begin{align*} G &= \{r/2^k : \text{$r \in \mathbb Z$, $0 \le r < 2^k$, $k = 0, 1, ...
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2answers
45 views

If a generated subgroup is cyclic

I would like to make a similar question to question "Exercise on generated subgroup": Let $G$ be a finite group and $H\leq G$, $H$ cyclic with $|H|=exp(G)$. If $x\in C_{G}(H)\smallsetminus H$, then ...
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1answer
34 views

Cycles of odd length: $\alpha^2=\beta^2 \implies \alpha=\beta$

Let $\alpha$ and $\beta$ be cycles of odd length (not disjoint). Prove that if $\alpha^2=\beta^2$, then $\alpha=\beta$. I need advice on how to approach this. I recognized that $\alpha,\beta$ are ...
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2answers
56 views

Proofread my work: Expressing generators of a cyclic group

The following question comes from Serge Lang's Undergraduate Algebra(pg. 26, 3rd edition). I just learnt the concept of groups and subgroups and I spent an hour or so on tackling part (b) of this ...
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1answer
70 views

Complement but not direct summand; Lam, Lectures on Modules and Rings, Example 6.17(5)

Let $S$ be a submodule of an $R$-module $M$. A submodule $C⊆M$ is said to be a complement to $S$ (in $M$) if $C$ is maximal with respect to the property that $C∩S=0$. (This does exist by Zorn Lemma.) ...
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1answer
72 views

When p groups are cyclic?

Let $|G|=p^n$ and have only one subgroup of order $p^{(n-1)}$.Then G is cyclic.I am trying it in many ways bt get nothing. What I get :The unique subgroup is normal in $G$, Center meets ...
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27 views

Determining all homomorphisms from $\mathbb Z_m$ to $\mathbb Z_n$ ? [duplicate]

What are all homomorphisms from $\mathbb Z_n$ to $\mathbb Z_n$ ? I know about all automorphisms but am not clear about all homomorphisms ; are there a total of $n$ homomorphisms ? In general , what ...
0
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1answer
36 views

Subgroups and cyclic groups [duplicate]

Suppose a group $G$ has no proper subgroups (that is, the only subgroup of $G$ is $G$ itself and the trivial subgroup $\{e\}$. Show that $G$ is cyclic.
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1answer
51 views

Is $\cos \frac{5\pi}{8}+i\sin\frac{5\pi}{8}$ in $U_8$, the multiplicative group of the $8$th roots of unity in $\mathbb{C}$?

I encountered this problem while reading Fraleigh, A First Course in Abstract Algebra, 4/e. (p.60 #19) Let $U_8$ be the multiplicative group of the $8$th roots of unity in $\mathbb{C}$. The question ...
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1answer
46 views

Left Cosets of Cyclic Subgroup

Question from a GRE Math book that I'm having trouble understanding: Find the number of left cosets of the cyclic subgroup generated by (1, 1) of $$Z_{2} \times Z_{4}$$ where Zn denotes the cyclic ...
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62 views

$G=\{f_n(x):n\in \mathbb{Z}\}$ is cyclic

Define $f_n(x)=x+n \;\;\forall n \in \mathbb{Z}$. Let $G=\{f_n:n\in \mathbb{Z}\}$. I proved that $G$ is a subgroup of $S_\mathbb{R}$ ($f_n$ is a permutation of $\mathbb{R}$), and now I am trying to ...
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23 views

Proving $|G|=pq$ and $p>q$ , $q$ does not divide $p-1$ $\implies$ $G$ is cyclic , without using Cauchy's and Sylow's theorems [duplicate]

Without using Cauchy's or Sylow's theorems , can we give a proof of the result that "If $ p,q$ are primes such that $p>q$ and $q$ does not divide $p-1$ , then any group of order $pq$ is cyclic " ? ...
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408 views

Does every group have a 'cyclization'?

Here's the question: Does every group have a 'cyclization'? That is, let $G$ be a group. Does there necessarily exist a cyclic group $C$ and a surjective homomorphism $\varphi:G\rightarrow C$ such ...
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1answer
68 views

A question on cyclic group with finite order

I have trouble proving the following statement: Suppose that $H$ is a finite group with order $n$ and $e$ is the identity element of $H$. For an arbitrary positive integer $d$ satisfing ...
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1answer
40 views

Rational group algebras and maximal orders

Let $G$ be a finite group, and $\mathbb{Q}[G]$ be the rational group algebra. Then the group ring $\mathbb{Z}[G]$ is an order in $\mathbb{Q}[G]$, but is not in general a maximal order. What can we ...
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23 views

Prove identity involving alternating groups

Prove the following identity: where $I_{A_n}(x_1,...,x_n)$ is a cyclic index the natural action of the alternating group $A_n$ on the set ${1,...,N}$ (assuming that $I_{A_0} = 1$).
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31 views

Cyclic groups: find power

Given the group $\mathbb Z_7$ and the generator $3$, we know the values generated are $a^0=3^0\pmod{7}=1$ $a^1=3^1\pmod{7}=3$ $a^2=3^2\pmod{7}=2$ $a^3=3^3\pmod{7}=6$ $a^4=3^4\pmod{7}=4$ ...
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32 views

If the automorphism group of a group is cyclic, then the group is commutative [duplicate]

Let $G$ be a group and the $Aut(G)$ group is cyclic $\Rightarrow$ the group $G$ is commutative. I looked at the homomorphism $\varphi : G \rightarrow Aut(G) \ g \mapsto (x \mapsto gxg^{-1})$. Let ...
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2answers
100 views

Show that $Z\times Z$ is not cyclic… [duplicate]

The full problem is as stated in the title. I am here to check if this is a valid proof. I thought it would be easiest using Linear Algebra. Recall that an infinite cyclic group is isomorphic to ...