All Questions
Tagged with group-theory cyclic-groups
1,728 questions
1
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G is a cyclic p-group iff every two subgroups have non-trivial intersection
Let G be a finite abelian group. Prove that G is a cyclic p-group if and only if for every two subgroups $H, F\le G$, we have $H\cap F\neq \{e\}$.
I came up with one implication ($\Rightarrow$), and ...
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2
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67
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Let $G$ be a non Abelian group for which it holds that all its subgroups $H$, $H\neq G$ are cyclic.
Let $G$ be a non Abelian group for which it holds that all its subgroups $H$, $H\neq G$ are cyclic.
(a) Find an example of such finite group $G$.
(b) Prove that every such group $G$ is generated with ...
user1489100
12
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6
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904
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Finite subgroups of multiplicative complex numbers.
In "A Course in Modern Mathematical Physics" as part of problem 2.1, Szekeres asks: "Find all finite subgroups of the multiplicative complex numbers $\dot{\mathbb{C}}$."
The ...
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1
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1
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Embedding of a cyclic group of order $n=2^r$ inside $A_{n+1}$
I'm looking for a proof regarding the following question.
If $r\geq1$ be a positive integer and $n=2^r$ then prove that a cyclic group of order n cannot be embedded inside the Alternating group $A_{n+...
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1
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1
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63
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Let $G$ and $H$ be cyclic groups. Prove: if $G$ is infinite group, then $G \times H$ is cyclic iff $|H|=1$. [duplicate]
Let $G$ and $H$ be cyclic groups. Prove: if $G$ is infinite group, then $G \times H$ is cyclic iff $|H|=1$.
Firstly from right to left I observe $H=\{e\}$. I want to prove that there exists $(a,e), a\...
user1483243
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5
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How to generate $0$ from $1$ and $-1$ in group of integers under addition operation?
I was following the Algebra book by Gallian. It mentioned that
"The set of integers $\mathbb{Z}$ under ordinary addition is cyclic. Both $1$ and $-1$ are generators".
This part is easy to ...
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Is there a standard term for a generator of maximal cyclic subgroup?
Is there a standard term for a group element that generates a maximal cyclic subgroup? Here "cyclic subgroup" means a subgroup generated by one element, and could be infinite. "Maximal ...
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$H$ abelian, finite group such that $\forall p$ prime dividing $|H|$, there exists exactly $p-1$ elements of order $p \Rightarrow H$ is cyclic.
I've had some trouble with this problem since I can't seem to know how to continue from certain point of my proof.
First of all, using Cauchy's theorem, we can assure that for every $p\mid |H|$ that ...
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What are the group extensions of $\mathbb{Z}_2$ by $U(1)$?
Given the group extension $1\rightarrow U(1)\overset{f}{\rightarrow}G\overset{g}{\rightarrow}\mathbb{Z}_2\rightarrow 1$, what are the possibilities for $G$?
I understand how to solve this for $1\...
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Finding all of the subgroups of order $p^r$ in $\mathbb Z _ {p ^ r} \times \mathbb Z _ {p ^ r}$
This question addresses how to find the cyclic subgroups of order $p^r$ in $\mathbb Z _ {p^r} \times \mathbb Z_{p^r}$. I wonder if it is possible to find all subgroups with two generators via the ...
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non-normal subgroups of a finite $p$-group
throughout studying a paper about finite $p$-groups,
I have a question as follows,
Let $G$ be a finite non-abelian $p$-group, where $p$ is odd and $Z(G)$ is cyclic.
($Z(G)$ denotes the center of $G$). ...
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2
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2
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93
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which polynomial has cyclic Galois group of order $n$ over $\mathbb{Q}$
It is known that the splitting field of $x^{p^n}-x$ over $\mathbb{F}_p$ is $\mathbf{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)\cong\mathbb{Z}/n\mathbb{Z}$ and the splitting field of $\Phi_n(x)$ over $\mathbb{...
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1
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Number of distinct decompositions of $Z/(p) \oplus Z/(p)$ in a direct sum of two proper subgroups
I have read some similar questions and answers but it was not easy for me to understand them. So I ask you to check my solution.
The question is to find the number of distinct decompositions of $Z/(p)...
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1
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Prove $C_2\times C_3$ is isomorphic to $C_6$ [duplicate]
I am struggling to understand whether or not this proof is legitimate, here is my attempt:
Prove that C2 × C3 is isomorphic to C6
Define $\varphi:C_6\rightarrow C_2\times C_3$ by the mapping $\varphi(...
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1
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88
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A question about cyclic groups
Apologize if it is elementary, but here it goes.
I am reading Basic Algebra I second edition from Nathan Jacobson and in page 95 he is trying to prove that if there is no isomorphism with the map ...
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1
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Describe all non-isomorphic groups of order $57$
Describe all non-isomorphic groups of order $57$, such that for each of them you write down its generators and the connections between them.
Attempt: 57 is the product of two primes, specifically $57 =...
user1345542
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1
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90
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How to prove that all elements inside a cycle of a cyclic group are different from each other [closed]
Let $G$ be a finite cyclic group $(G, \circ)$ and $a \in G$:
$$
\langle a \rangle = \{a^z : z \in \mathbb{Z}\},
$$
and $\operatorname{ord}(a) = \min\{a^n : n \in \mathbb{N}_+\}$, i.e., the smallest ...
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1
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93
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Why is the order of an element equal to the order of the group it generates? [duplicate]
I've found this post Prove the order of an element is the order of the group but this does not help me.
Let G be a finite group (G,$\circ$) and a $\in$ G.
$ \langle a \rangle $ = {$a^z$ : z $\in \...
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3
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122
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showing cyclic groups are abelian when the operation is not addtion or multiplication using fancy example
To begin, if my question is foolish, i apologize for it, but i am trying to learn by myself, and some questions exist in my head.
I am trying to understand why every cyclic group is abelian. It is ...
user1347971
2
votes
1
answer
100
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An infinite cyclic group has infinitely many irreducible real representations.
I'm trying to show that an infinite cyclic group $G=\langle g\rangle$ has infinitely many non-equivalent irreducible representations over $\mathbb{R}$. I have in mind the following argument, for which ...
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0
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Clarifying question regarding Eulers totient function and the order of ${Z_{P}}^{\times}$ for p prime.
So on this webpage: https://crypto.stanford.edu/pbc/notes/numbertheory/gen.html I read that
Let p be prime, ${Z_{P}}^{\times}$ contains exactly $\phi (p-1)$ generators.
Additionally, in this post:...
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0
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50
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A proper subgroup of rational numbers which is not cyclic [duplicate]
How do we prove that the dyadic rationals $\{ \frac{a}{2^n} : a,n \in \mathbb{Z} \}$ is a "proper" subgroup of $\mathbb{Q}$?
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43
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Suppose that $p,q$ are primes,What's the necessary and sufficient condition for the group of order pq to be cyclic. [duplicate]
The question came from an example on the book "Contemporary Abstract Algebra" (9th Edition) in page 203.The orginal problem is to prove that every group of order 35 is cyclic.I wondered if ...
user1325170
0
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1
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86
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Proving $o(H\cap K)=\gcd(o(H),o(K))$. [closed]
Let $H$ and $K$ be subgroups of a finite cyclic group $G,$ then $|H \cap K| = \gcd(|H|,|K|)$.*The same question is already asked in this plateform here.
So this is indeed a duplicate question but i am ...
user1313860
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Any group of order 33 is cyclic **using conjugation action** of subgroup of G on G) [duplicate]
Question Prove that any group G of order 33 is cyclic, by considering the conjugation action of a subgroup of G on G.
This is a repeat question, I really do not understand how I am supposed to use ...
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1
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74
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How many homomorphisms $C_9 \rightarrow S_4$ are there? [duplicate]
How many homomorphisms $C_9 \rightarrow S_4$ are there?
[I want to use the fact that the order of $\phi(g)$ divides the order of $g$ and somehow work from there (I also know that if $\phi$ is ...
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38
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Number of fixed elements of $a\mapsto a^r$ in the group $\mathbb{Z}_{pq}^*$ [duplicate]
Suppose $n=pq$, with $p$ and $q$ distinct primes and $r$ is coprime to $\phi(n)=(p-1)(q-1)$. Consider the map $f:\mathbb{Z}_n^*\to \mathbb{Z}_n^*$ given by $a\mapsto a^r$, where
Then show that the ...
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68
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Proof that a group with only characteristic subgroups is abelian with locally cyclic torsion subgroup
I'm trying to understand a proof by Cutolo, Smith and Wiegold that a group $G$ all of whose subgroups are characteristic is necessarily abelian and has locally cyclic torsion subgroup.
I think I ...
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1
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57
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If $X$ is a subnormal in $G$ with $[a, X] = 1 = [a, C_G(X)]$, then $[a, G] = G$
Let $G$ be a $\pi$-group and $a$ $\pi'$-element that acts on $G$. If $X$ is a subnormal subgroup of $G$ with $[a, X] = 1 = [a, C_G(X)]$, then $[a, G] = G$.
Proof
Let $X = X_0 \unlhd X_1 \unlhd \...
user1175180
1
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0
answers
98
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About semi-direct product of two cyclic groups
The following question is related to seeing semi-direct products as subgroups:
Let $G$ be a non-nilpotent group and $U = \langle x \rangle$ be a cyclic characteristic subgroup of the Fitting subgroup $...
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2
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63
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Lifting map from finite cyclic group to integers
Suppose $\phi:G \to \mathbb Z/n\mathbb Z$ is a group quotient with finite cyclic image. Under what conditions can $\phi$ be lifted to a homomorphism $\tilde \phi:G \to \mathbb Z$?
Specifically I'm ...
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1
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What does the book mean? Confusion over element $a$ in theorem $\langle a^k\rangle=\langle a^{(n,k)}\rangle$
This question may seem silly, but it is important for me.
My background: Non-native english speaker and first year college student working abstract algebra by myself.
Theorem: Let $a$ be an element ...
user1307137
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3
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But what is with the other cyclic groups? Doesn't one also have to consider them? [closed]
I'm currently reading a textbook about abstract algebra.
There is a proof that every subgroup of a cyclic group is cyclic.
This proof is using the fact as every proof I have found on the Internet ...
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-1
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1
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51
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generalized version of definition of cyclic groups and its abelian property [closed]
I am beginner in group theory and I have learned cyclic groups yet. When I look at the books for definition, it is said that
Let $G$ be a cyclic group and let a be a generator of $G$ so that $$G= \...
user1304425
5
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1
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Schur's Multiplier exercise (Problem 5A.7 Isaacs' Finite Group Theory)
I have a question about the following problem [Finite Group Theory, Martin Isaacs, Chapter 5]:
Let $ B $ and $ C $ be cyclic subgroups of a finite group G, and suppose that $ BC = G $ and $ B \cap C &...
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1
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86
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$p$-group with a cyclic subgroup
Throughout studying a paper about finite $p$-groups,
I have the following question
Let $G$ be a finite $p$-group with nilpotency class 3 and $\gamma_i(G)$ denote
the i'th term of the lower central ...
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-1
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3
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194
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Non-cyclic subgroup of order 4 in non-dihedral group
A group $G$ has sixteen elements:
$$\{e, r, r^2, \dots , r^7, s, rs, r^2s, \dots , r^7s\},$$
where $r$ and $s$ satisfy the relations $r^8 = e, s^2 = e, sr = r^3s$.
(Note that $G$ is not a dihedral ...
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4
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1
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95
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An abelian group of order 35 must be cyclic [duplicate]
I'm reading through Contemporary Abstract Algebra by Joseph A. Galian and I stumbled upon the following problem:
Suppose $G$ is an Abelian group of order $35$ and every element of $G$ satisfies $x^{...
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2
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2
answers
80
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Determine the number of group homomorphisms $f:\mathbb Z_{63}\to\mathbb Z_{147}$ with $|image(f)|=7$.
Solution is apparently 6.
Current attempt, although, I am not sure whether this the right approach:
Let $f(x):=x$. We require $f(63)=63x\equiv 0 \mod 147$; that is, $63x=147k\iff 7\mid x$. So we have ...
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1
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1
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Understanding Why a Product of Cyclic Groups with Non-Coprime Orders is not Cyclic
Let $G_1, G_2, \ldots, G_t$ be finite cyclic groups, and define $G = G_1 \times G_2 \times \ldots \times G_t$. Let $n_j = |G_j|$, such that $|G| = \prod_{j=1}^{t} n_j$.
For part (a) of the problem, we ...
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226
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Why is $\langle 3 \rangle$ a generator of $(\mathbb{Z}_7^*, \cdot)$?
This link here shows that $\langle 3 \rangle$ is a generator for the given group by brute force method, that is trial and error. I was curious as to how to justify this using the theorem.
According to ...
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0
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84
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Group of order 1001 is cyclic. (Basic Group Theory Solution) [duplicate]
I am trying to prove that if $G$ is a group of order $1001$, then $G$ has normal subgroups of order $7$, $11$, and $13$, and thus $G$ is cyclic. I know that by Cauchy's Theorem, there exist elements $...
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1
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84
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How to find subgroups of direct product of cyclic groups [duplicate]
Find all subgroups of $\mathbb{Z_2} \times \mathbb{Z_5} \times \mathbb{Z_7}$.
I know that this would be isomorphic to $\mathbb{Z_{70}}$, so that I would be looking for $8$ subgroups because $70$ has $...
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0
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1
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131
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Reviewing a proof to show that a subgroup of a cyclic group is also cyclic [closed]
I want to prove the following statement:
Let $H$ be a subgroup of a cyclic group $G$, here, finite. Then $H$ is also cyclic.
My question is about an attempt that I include below; in particular, ...
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-1
votes
1
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104
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Is there any External Direct Product which is cyclic but it is the product of two non-cyclic groups? [closed]
Is there any External Direct Product which is cyclic but it is the product of two non-cyclic groups?
I know that for an EDP, say $G\times G'$ to be cyclic the $\gcd(o(G),o(G'))=1$.
But I am unable to ...
2
votes
1
answer
124
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show G has a faithful representation of degree 1 over $\mathbb{C}$ iff it is cyclic
Show that a finite group G has a faithful representation of degree 1 over $\mathbb{C}$ iff G is cyclic.
I saw the following related post, but I'm not sure if I fully understand it: Faithful ...
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3
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180
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What is a group and its operation?
I am learning group theory by myself and has not reached cyclic groups yet but as I have read cyclic groups are group generated by a single element and is denoted as $G = \langle x, *\rangle$ where $*$...
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0
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94
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Prove that when $|H| = \infty$, $\langle x^a \rangle \neq \langle x^b \rangle$ and $\langle x^m \rangle = \langle x^{|m|} \rangle$
I want to prove Theorem 7(2) in section 2.3 from Abstract Algebra 3rd Edition by Dummit & Foote. This theorem states:
Let $H = \langle x \rangle$ be a cyclic group. If $|H| = \infty$, then for ...
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2
votes
0
answers
74
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Invariants of the Hyperoctahedral group
Apologies in advance for what I am asking might be too trivial, I am not a mathematician.
I have a function $V(x_1,\dots,x_n)$ that could have, at most, a hyperoctahedral (signed permutations) ...
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0
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65
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Invariant factors and elementary divisors of quotient group
I need help solving the following problem:
Let $G = \Bbb{Z}_9 \times \Bbb{Z}_9 \times \Bbb{Z}_9$ be the product of three cyclic groups of order 9. Give invariant factors and elementary divisors of ...
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