2
votes
2answers
69 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 ...
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
204 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
51 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 $G$?
2
votes
6answers
71 views

A question on cyclic group

I have no trouble proving the following statement. Let $G$ be a cyclic group of order $n$ and let $k$ be an integer relatively prime to $n$. Prove the map $x \mapsto x^k$ is surjective. It is clear by ...
4
votes
2answers
41 views

What is this notation? Cyclic group $\mathbb{Z}^*_8$

$\mathbb{Z}^*_8$ As I understand it - $\mathbb{Z}_8$ is the group of integers under addition modulo 8. So am I correct in thinking its elements are: $\{0,1,2,3,4,5,6,7\}$? I thought the $*$ meant ...
1
vote
2answers
22 views

if $f$ is of prime order, then can the orbit of $s$ under $f$ have one element?

if $f\in A(S)$ has order $p$, $p$ a prime, show that for every $s\in S$ the orbit of $s$ under $f$ has one or $p$ elements.* Since the cyclic group generated by $f$ has $p$ elements, therefore ...
1
vote
1answer
55 views

How to count generators for a cyclic group

Show that there are $\varphi (n)$ generators for a cyclic group $G$ of order $n$. Give their form explicitly. Here $\varphi (n)$ is the Euler's function. I don't know what to do, please help.
0
votes
1answer
35 views

Abstract Algebra: prove it is cyclic

I have question in referring to below link. Question. Suppose if I have [M:K]=2 and I know that K is subset of M. M:=$\mathbb{Z}_2[x]/(f(x))$ where f(x)=x$^4$+x+1. Then how this will be cyclic? I ...
2
votes
2answers
92 views

How many subgroups of $\Bbb{Z}_4 \times \Bbb{Z}_6$?

I have been trying to calculate the number of subgroups of the direct cross product $\Bbb{Z}_4 \times \Bbb{Z}_6.$ Using Goursat's Theorem, I can calculate 16. Here's the info: Goursat's Theorem: Let ...
2
votes
2answers
93 views

group theory proof

Let $G=\langle t \rangle$ be a cyclic group with $\text{ord}(t)=n$. I want to show that for all $d|n$ it holds that $$\left\lbrace s \in G ;\text{ord}(s)=d \right\rbrace=\left\lbrace t^{\frac{n}{d}k} ...
1
vote
1answer
26 views

Does this condition gaurantee the cyclicity of a finite abelian group?

Let $G$ be a finite abelian group in which there are at most $n$ solutions of the equation $x^n = e$ for each posivite integer $n$. How to determine if $G$ is cyclic or not?
0
votes
1answer
47 views

Which elements of this cyclic group would generate it?

Let $n$ be a given arbitrary positive integer, and let $U_n$ denote the group of all the positive integers less than $n$ and relatively prime to $n$ under multiplication mod $n$. Then for which values ...
2
votes
1answer
38 views

Abstract Algebra: Cosets

Find all of the distinct left cosets of <4> in Z18 and all the cosets of <4> in the subgroup <2> of Z18. So The distinct left cosets of <4> in Z18 are 0 + <4> and 1 + <4>. Do I ...
1
vote
1answer
35 views

Regarding the order of elements in a factor group

If I have understood it correctly, a factor group consists of all cosets of a subgroup $H$ of $G$. Since it is a requirement that $H$ is normal, left and right cosets are equivalent. I am asked to ...
2
votes
2answers
67 views

Morphisms between $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}/m\mathbb{Z}$

I'm trying to determinate how many morphisms of groupes exist between $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}/m\mathbb{Z}$ for $n,m\in\mathbb{N}$. I know a morphism is determinated by the image of ...
2
votes
1answer
60 views

Nontrivial Homomorphism(s) from $\mathbb{Z_3}$ to $S_3$ - Fraleigh p. 134 13.37

Reference: http://users.humboldt.edu/pgoetz/Homework%20Solutions/Math%20343/hwsome number 1 to 17 that I forgotsolns.pdf There are exactly two nontrivial ...
2
votes
2answers
150 views

When is the automorphism group $\text{Aut }G$ cyclic?

Let $G$ be a finite group. Under which conditions on $G$ is the automorphism group $\text{Aut }G$ cyclic? More precisely, does $G$ is abelian or $G$ is cyclic implies $\text{Aut }G$ is cyclic?
2
votes
2answers
73 views

Fast way of calculating the order of an element in $\mathbb{Z}_n$?

Is there a fast way of calculating the order of an element in $\mathbb{Z}_n$? If i'm asked to calculate the order of $12 \in \mathbb{Z}_{22}$ I just sit there adding $12$ to itself and seeing if the ...
0
votes
1answer
36 views

If $\sigma\in S_n$ is of length $n$, then $\sigma$ generates its centralizer

Let $S_n$ and $C_G(\sigma)$ denote the symmetric goup and the centralizer of $\sigma\in S_n$, respectively. I want to show: If $\sigma$ is of length $n$, then $C_G(\sigma)=\langle\sigma\rangle$, i.e. ...
1
vote
1answer
63 views

If $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$.

Let $G$ be finite abelian group and $\hat G$ be its character group. I need hint proving that if $a\in G$ and $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$ (the identity element). I can prove it ...
2
votes
1answer
62 views

subgroups of finite cyclic group

Let $G=(g)$ be a finite cyclic group generated by $g$ with $|G|=n$, and let $d \in \mathbb{N}$ with $d|n$, then an unique subgroup $H$ of $G$ with $|H|=d$ exists. Proof of existence: $\exists m \in ...
-3
votes
3answers
94 views

Let $G = \{1, a, b, c\}$ be a group of order 4…Exist two groups of order $4$.

Let $ G = \{1, a, b, c\}$ be a group of order 4. Show that, if $G$ is cyclic $G \cong \mathbb Z_4,$ and if $G$ is not cyclic then $G \cong K_4.$ It now follows that there are only 2 groups of order ...
0
votes
2answers
92 views

quotient group Z/3Z equated to 0,1,2

This a followup question on the first answer to this post: Why the term and the concept of quotient group? In the first answer, Lahtonen says that for the quotient group Z/10Z, one can "equate 9 ...
1
vote
4answers
131 views

Semi-direct product of groups with one of them cyclic

Let $K$ be a cyclic group. Let $\phi,\psi: K\rightarrow Aut(H)$ be group homomorphisms such that there exists $\zeta\in Aut(H)$ satisfying $\phi(K)=\zeta \psi(K)\zeta^{-1}$. Then can we prove ...
0
votes
2answers
165 views

surjective homomorphisms between cyclic groups (wrong question)

This statement does not hold. Let $C, D$ be cyclic groups and $f_1,f_2:C\rightarrow D$ be surjective homomorphisms. Show that there exists $\xi: C\rightarrow C$ such that $f_2=f_1\circ \xi$. My ...
4
votes
2answers
86 views

Suppose a unique a generates a cyclic subgroup of order. Show ax = xa. - Fraleigh p. 67 6.50

(1.) I don't understand above. How do you magically envisage and envision to let $b = xax^{-1}$? What I did was to start from the answer and see if I can get a chain of equivalences. $ax = xa ...
5
votes
4answers
223 views

Show that if $ab$ has finite order $n$, then $ba$ also has order $n$. - Fraleigh p. 47 6.46.

This solution is from here and yahoo. Given $a,b$ elements of $G$, and $ab$ has finite order $n$. Hence $\color{magenta}{|ab| = n} \iff (ab)^n = e$. Need to show $n$ is the smallest positive integer ...
1
vote
2answers
59 views

Order of a cyclic group with a single proper subgroup of order 7

Let $G$ be a cyclic group with its only proper subgroup of order 7. Find out the order of the group. let the subgroup of order 7 be denoted by H. since 7 is prime H is cyclic. Now if G = then ...
5
votes
3answers
181 views

Intuition and Tricks - Hard Overcomplex Proof - Order of Subgroup of Cyclic Subgroup - Fraleigh p. 64 Theorem 6.14

Update Dec. 28 2013. See a stronger result and easier proof here. I didn't find it until after I posted this. This isn't a duplicate. Proof is based on ProofWiki. But I leave out the redundant $a$. ...
1
vote
0answers
58 views

Proving a theorem at groups theory [duplicate]

$G$ is a finite group. Let assume that for every $m\in \mathbb{N}$ there are maximum $m$ elements s.t. $x^m=e\;$. $(x\in G)$ I need to prove that $G$ is cyclic group. I need to use the fact that: ...
4
votes
0answers
114 views

$(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ is a cyclic group

I would like to prove that the group $(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ of the invertible elements of $\mathbb{Z}/p^r\mathbb{Z}$ with $p>2$ prime and $r>0$ is cyclic. My text suggests to ...
-1
votes
2answers
215 views

Prove that in any group an element and its inverse have same order. [closed]

Prove that an element and its inverse have same order in any group.
1
vote
3answers
117 views

Trouble understanding this proof about cyclic groups of order $p^n$

This is a theorem / proof from Rotman, and I am having a little trouble following it. I've reproduced the theorem and proof (proof is not verbatim) below. Theorem: Let $p$ be a prime. A group ...
2
votes
1answer
234 views

Any group of order $n$ satisfying $\gcd (n, \varphi(n)) =1$ is cyclic

Sorry for the last mistaken problem I just posted. Now I know that only having the order being odd square free is not enough for a group to be cyclic. Here's the complete problem which the main goal ...
3
votes
3answers
194 views

Any group of odd square free order is cyclic

So this problem is actually mistaken, the condition should really be that a group of order $n$ with $\gcd(n,\varphi(n))=1$, not just being odd square free. Since there exists a group of order 21 which ...
1
vote
1answer
49 views

Question about $e$ element at $\mathbb{Z}_{n}$

At group $\mathbb{Z}_{n}$, $e=0$? I assume that is true but I just what to know if I'm right. Because for every $a\in \mathbb{Z}_{n}, a^0=a\cdot 0=0$. Thank you!
1
vote
1answer
24 views

coproduct of cyclic groups

If G is a nilpotent group satisfying the maximal condition on normal subgroups then each subgroup H of G is finitely generated. This is the statement I believe it is true. However I do not know how ...
3
votes
1answer
70 views

The number of cyclic subgroups of order 15 in $\mathbb{Z}_{30} \oplus \mathbb{Z}_{20}$

What is the number of cyclic subgroup of order 15 in $\mathbb{Z}_{30} \oplus \mathbb{Z}_{20}$? I have counted the number of elements with order 15 in $\mathbb{Z}_{30} \oplus \mathbb{Z}_{20}$ that ...
2
votes
2answers
136 views

How Many Cyclic Subgroups of order $10$ are there in $\mathbb{Z}_{100}\oplus\mathbb{Z}_{25}$

How Many Cyclic Subgroups of order $10$ are there in $\mathbb{Z}_{100}\oplus\mathbb{Z}_{25}$? I have calculated that there are $24$ elements of order $10$ I know that in a cyclic subgroup of order ...
4
votes
0answers
60 views

Proving a group is cyclic [duplicate]

Let $G$ be a group of order $pq$, where $p,q$ are primes, $p < q$ and $q≢1$ (mod $p$); how do we prove that $G$ is cyclic ? (I have no idea)
0
votes
1answer
40 views

Prove that $\langle a\rangle = G$, where $G$ a cyclic group of order 24, $a \in G$, $a^8 \ne e$, $a^{12} \ne e$.

Let $G$ a cyclic group, $|G| = 24$. Let $a\in G$, such that $a^8 \ne e$ and $a^{12}\ne e$. Prove that $\langle a \rangle = G$. So far i have found that $a^2,a^3,a^4,a^6 \ne e$. So to solve the ...
0
votes
0answers
70 views

Showing that $ U(2^n) $ is not a cyclic group for $ n \geq 3 $ [duplicate]

Could anyone please explain to me why $ U(2^n) $ is not a cyclic group for $ n \geq 3 $? I need help on this because I have an algebra exam tomorrow. Thanks!
0
votes
0answers
275 views

Is the multiplicative group of non-zero elements of a finite field cyclic? What are the generators of $(Z_p \setminus \{0\}$ for a prime $p$?

How to decide if the multiplicative group of non-zero elements of a finite field is cyclic or not? Based on our experience with $(Z_7 \setminus \{0\}, \cdot)$, which is generated by $3$ or $5$, and ...
1
vote
3answers
141 views

Suppose that $G$ is a finite cyclic group. Let $m=|G|$. Assume that $m\ge3$. Let $S=\{a\in G:|a|=m\}$. Prove that the cardinality of $S$ is even.

Yeah, NO IDEA WHERE TO GO FROM HERE. $|S|$ has to be equal to $2k$ with $k$ being a positive integer, but someone please offer a hint on how I can get started on this.
2
votes
1answer
136 views

At most one subgroup of every order dividing $\lvert G\rvert$ implies $G$ cyclic [closed]

Suppose we have a finite group $G$ of finite order $n$. For every $d\mid n$, $G$ has at most one subgroup of order $d$. Show that $G$ is cyclic.
0
votes
1answer
66 views

What is the necessary and sufficient condition for this Cartesian product to be a cyclic group?

Let $m_1$, $m_2$, $\ldots$, $m_n$ be positive integers, and let $Z_{m_i}$ denote the group $\{0, 1, 2, \ldots, m_{i}-1\}$ under addition modulo $m_i$, for each $i = 1,2, \ldots, n$. Then what is the ...
10
votes
2answers
261 views

Can we conclude that this group is cyclic? [duplicate]

Let $G$ be a finite group. If, for each positive integer $m$, the number of solutions of the equation $x^m = e$ in $G$, where $e$ is the identity element, is at most $m$, then can we conclude that $G$ ...
-1
votes
1answer
79 views

What is the relation between these two subgroups of a finite cyclic group?

Let $G$ be a finite cyclic group of order $n$ generated by $a$. If $k$, $m$ are integers such that gcd($n, m$) = gcd($n,k$), then what is the relationship between the subgroups generated by $a^k$ and ...
0
votes
2answers
100 views

For group $\mathbb{Z_{18}^*}$, how do I find all subgroups

In my textbook, there is a cyclic Group $G=\mathbb{Z_{18}^*}$ which has the elements $$\{1,5,7,11,13,17\}$$ And its subgroups are $U_1 = \{1\}$, $U_2 = \{1,17\}$ and $U_3 = \{1,7,13\}$ How did they ...