2
votes
1answer
35 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. ...
0
votes
1answer
74 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 ...
1
vote
1answer
49 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
0
votes
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 ...
1
vote
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 ...
2
votes
3answers
28 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 ...
2
votes
4answers
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: ...
-1
votes
0answers
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 ...
0
votes
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 ...
0
votes
3answers
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 ...
2
votes
2answers
48 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 $\ker f$. ...
5
votes
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 ...
1
vote
2answers
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 $d$ ...
0
votes
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 ...
0
votes
2answers
49 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 ...
-2
votes
0answers
39 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 ...
0
votes
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) = ...
-2
votes
2answers
57 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 ...
3
votes
2answers
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, ...
0
votes
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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
0answers
25 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 ...
3
votes
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 ...
1
vote
1answer
45 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 ...
1
vote
2answers
61 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 ...
1
vote
0answers
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 " ? ...
12
votes
4answers
407 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 ...
0
votes
0answers
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 $I_{A_0} = 1$).
0
votes
0answers
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 ...
1
vote
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 ...
1
vote
1answer
44 views

Show whether the following groups are cyclic or not…

Here's the full problem: Show whether the groups $G_{10},G_{7}$ are cyclic or not. If so, find their generators. $G_{10}$ and $G_7$ are the sets of invertible elements mod 10 and mod 7. So, I ...
2
votes
2answers
76 views

Cyclic subgroups of semidirect products

Let $H$ and $N$ be two finite groups and $\phi:H\to Aut(N)$ a homomorphism. Let $G=N\rtimes_\phi H$ and let $\pi:G\to G/N=H$ be the quotient map. Let $p$ be a prime which does not divide $|N|$ but ...
1
vote
2answers
68 views

What exactly does $\langle a,b \rangle$ mean? Where a and b are elements of a group G?

Does it mean either a or b will generate the whole of $\langle a,b \rangle$ or does it mean that some of the elements will be generated by a, some by b, and some by $a^rb^q$? The book I'm reading ...
2
votes
2answers
56 views

Are the only generators of a Cyclic Group $G=\langle g\rangle$, where $|g| = \infty$, $g$ and $g^{-1}$?

I'm self studying group theory, and this is a question in the textbook I've taken out, there is no answer given so I'm assuming that's because it's too simple to require one. I'm almost certain that ...
4
votes
3answers
102 views

Finding kernel of homomorphism $f:\mathbb Z \to S_8$ such that $f(1)=(1426)(257)$

Let $f:\mathbb Z \to S_8$ be a homomorphism such that $f(1)=(1426)(257)$ , then how to compute $\ker(f)$ and $f(20)$? I know that $f(n)=f^n(1)$ but this seems too tedious; please help
0
votes
2answers
76 views

Prove that if G is a cyclic group with more than 2 elements, then there always exists an isomorphism ϕ:G→G that is not the identity mapping [duplicate]

The full question is Prove that if G is a cyclic group with more than 2 elements, then there always exists an isomorphism ϕ:G→G that is not the identity mapping.I have no idea where to start?
3
votes
3answers
176 views

A persisting element in all subgroups.

Let $G$ be a finitely generated abelian group and $a$ be a nontrival element of $G$ contained in all nontrivial subgroups of $G$. Is $G$ necessarily cyclic?
-1
votes
3answers
66 views

$G$ is an abelian group of order a product of distinct primes $\implies G$ is cyclic?

If $G$ is an abelian group of order $p_1p_2...p_k$ , where $p_1,p_2,...,p_k$ are distinct primes , then is it true that $G$ is cyclic ?
4
votes
1answer
65 views

Another Presentation of Certain Cyclic Groups

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^n\rangle $$ is cyclic of order $3(n+1)$, for $n=0 \mod 3$ or $n= 1 \mod 3$, $n\ge 0$. This ...
4
votes
2answers
210 views

Cyclic Group Presentation

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^7\rangle $$ is cyclic of order 24. This presentation was obtained using the Todd-Coxeter ...
1
vote
1answer
83 views

Artin Algebra 2.8.3 “Does every group whose order is a power of a prime $p$ contains an element of order $p$?”

I'm not sure whether or not my answer and proof for this question are valid. Could you point out any flaw? Let $G$ be an arbitrary group, an arbitrary element of $G$ be $g$ and $|G|=p^n$. Since a ...
1
vote
1answer
53 views

What are all the automorphisms of a group of order $9$ generated by two elements?

Let $G$ be a group of order $9$ generated by two elements $a$ and $b$ such that $a^3 = b^3 =e$. How to determine all possible automorphisms of $G$?
0
votes
3answers
71 views

Cosets/ Cyclic group

Let $G=\langle a\rangle$ and $H=\langle a^2\rangle$. Find all the right cosets of $H$ in $G$. Additional info: I understand that a right coset of $H$ in $G$ is of the form $Ha=\{ha:h \in H\}$. But I ...
0
votes
0answers
26 views

Be $G$ a group of order $|G|=2p$ where $p$ is a prime number odd, prove that either $ G $ is cyclic or dihedral $G\simeq D_p$ the group of order 2$p$. [duplicate]

Be $G$ a group of order $|G|=2p$ where $p$ is a prime number odd, prove that either $ G $ is cyclic or dihedral $G\simeq D_p$ the group of order 2$p$. I thought the question a little difficult ...
0
votes
0answers
65 views

Free groups vs. free abelian groups

I'm trying to solve this question in page 74 of Hungerford's book: A free abelian group is a free group (Section I.9) if and only if it is cyclic. I have no idea how to proceed, a solution or a ...
4
votes
5answers
379 views

Prove any subgroup of a cyclic group is cyclic.

was just wondering if this is a valid proof for the aforementioned question? I am quite confident that it isn't, but not exactly sure why. Maybe I am missing the point of proofs by induction ...
1
vote
2answers
38 views

Why $(\mathbb{Z}/N\mathbb{Z})^{\times}[2]$ is of order $2$?

Why if $(\mathbb{Z}/N\mathbb{Z})^{\times}$ is cyclic, the group of his elements of order dividing $2$ is of order 2?
2
votes
3answers
73 views

Find a generator of the multiplicative group of $\mathbb{Z}/23\mathbb{Z}$ as a cyclic group

I need to find a generator of the multiplicative group of $\mathbb{Z}/23\mathbb{Z}$ as a cyclic group. Since $\mathbb{Z}/23\mathbb{Z}$ only has $23$ elements and ord$(x)$ where $x$ is a generator ...
1
vote
1answer
57 views

Show that a subset of the $2 \times 2$ matrices is an infinite cyclic group

Let $M$ Denote the set of 2x2-matrices of the form $$A=\pmatrix{1&m\\0&1}$$ where the entries are integers. Show that $M$, with respect to matrix multiplicaation, is an infinite cyclic ...