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Questions tagged [group-theory]

A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory is the study of groups.

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Burnside convolution

Let $G$ be a group. Say that an orbit is a nonempty transitive $G$-set. Let $\Xi$ be a set of finite orbits such that each finite orbit is isomorphic to exactly one element of $\Xi$. If $X,Y,Z\in\Xi$,...
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20 views

Infinite Abelian Groups: Unnecessary Hypothesis in Kaplansky's Lemma 7

Looking for a validation of a proof of a stronger statement. The essential argument that needs review is the final paragraph. This would obviate the need for a much more complicated argument I gave ...
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1answer
22 views

How many group homomorphisms are there from $(\mathbb{Q}^+, . )$ to $(\mathbb{Z}_m,+)$

Ofcourse there is the trivial zero homomorphism. I think there is no more homomorphisms there but I am failing to prove it.
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1answer
48 views

Elliptic curve addition: why does it work in any field?

Supposing I have an elliptic curve E(K): y^2 = x^3 + Ax + B with char(K) != 2,3, why do the formulas for EC addition work in any field. The formulas make sense in ℝ, for example we calculate the slope ...
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1answer
20 views

Does there exist a verbally simple group, which is not characteristically simple?

Does there exist a verbally simple group, which is not characteristically simple? A characteristically simple group is a group without non-trivial proper characteristic subgroups, a verbally simple ...
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1answer
26 views

Exact sequence of groups: proof of injectivity

There must be a duplicate being the question very introductory, but I was not able to find it. We have the following diagram $$\begin{array}{ccccccccc} 1&\to& H &\to & G &\...
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2answers
48 views

How to show that $H$ is normal in $G$?

Let $G$ be a group of order $pq$, where $p$ and $q$ are primes and $p>q$. Ler $a\in G$ be of order $p$ and $H=\big<a: a^p=1\big>.$ Then $H$ is normal in $G$. I know that $H$ will be normal ...
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7 views

What is the action of a primitive permutation group of type SD?

My goal is to show that $Alt(5) \wr Sym(3)$ is a primitive permutation group of type SD. Let $G$ be a primitive permutation group, $N$ a minimal normal subgroup, and $H$ a point stabilizer (so $NH =...
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2answers
32 views

Confusion about a proof about subgroups of dihedral groups

This article shows that every subgroup of $D_n = \langle r, s \rangle$ is cyclic or dihedral. Theorem 3.1. Every subgroup of $D_n = \langle r, s \rangle$ is cyclic or dihedral. A complete listing ...
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0answers
18 views

Group proof/disproof v2 [duplicate]

How do I prove or disprove that, for a group $G$, if $a,b\in{}G$ and $aa_0=b$, then $a_0\in{}G$? Edit: I sillily said $b\in{}G$ instead of $a_0\in{}G$ originally. This is the reposted version written ...
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1answer
24 views

Show the Dihedral Group $D_n$ is generated by rotations and reflection along the x axis.

I'm having problems understanding the excersice: E) Define $D_n$ as the group of symmetries of a regular n-gon. Name the vertices $V=\{V_0,V_1,...,V_{n-1}\}$ so that $$V_{k}=\exp({i\cdot\dfrac{2\pi k}{...
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2answers
32 views

Group proof/disproof [on hold]

How do I prove or disprove that, for a group $G$, if $a,b\in{}G$ and $aa_0=b$, then $a_0\in{}G$? Edit: I sillily said $b\in{}G$ instead of $a_0\in{}G$ originally.
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2answers
40 views

Prove if the the groups $A_4$ and $S_3 \times \Bbb Z_2$ are or not are isomorphic

I'm trying to check if the groups $A_4$ and $S_3 \times \Bbb Z_2$ are or not isomorphic. How can I check if they are? I'm trying to understand how can I generally prove an isomorphism with this kind ...
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0answers
29 views

Determine the number of cycles of the form $(\underline{6})(\underline{2})(\underline{2})$ in $S_9$

Consider a product of cycles of the form $(\underline{6})(\underline{2})(\underline{2})$ in $S_9$ where $(\underline{i})$ is a cycle with length $i$. I want to determine the number of such cycles. ...
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1answer
21 views

Direct product decomposition of the group of complex roots of unity

I'm studying $p$-adic numbers (Robert's "A course in $p$-adic analysis) and, at page 41, the author states that, for every prime $p$, the group $\mu$ of all complex roots of unity has a direct product ...
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2answers
31 views

Homorphism between groups and generators

Let $f: G \rightarrow H$ be a homomorphism of finite groups. Since $G$ is finitely generated, $G = \langle x_1 , ... , x_n \rangle$. Is it then true that $H = \langle f(x_1), ... , f(x_n) \rangle$? ...
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2answers
29 views

Prove that family of subsets is the set of cosets of some subroup of group

Exercise Let $G$ be a finite group and suppose that $\mathbf{C}$ is a family of subsets of a group $G$ which for a partition of $G$. Further suppose that $gC \in \mathbf{C}$ for any $g \in G$ and $C \...
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0answers
31 views

Proof that the automorphisms of a monoid form a group [on hold]

Prove that the set of all automorphisms of a monoid M forms a group under composition. I know that an automorphism of a monoid M is an isomorphism from M to itself but I don't know how to use this ...
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1answer
24 views

Group representation preserving finitely many generators

Let $\rho : G \rightarrow GL_n(\mathbb{C})$ be a representation of $G$. If $G$ is finitely generated as a group, does that mean that $im(\rho) \leq GL_n(\mathbb{C}) $ is finitely generated? Because, ...
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25 views

Abelian subgroups of a simple group

Let $G$ be a finite non-abelian simple group, and $A \leqslant G$ be an abelian subgroup. How large $A$ can be? There exists any bound of the type $|A| \leq |G|^r$ for some $r<1$? How can I prove ...
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1answer
29 views

Pólya-Burnside Method of Enumeration with Necklace

Use the Pólya-Burnside Method of Enumeration or just the Burnside theorem to find the number of different types of circular necklaces that could be made from the following beads (assuming all are used ...
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1answer
33 views

Nontrivial normal subgroup that doesn't contain commutator subgroup

Can there be non-trivial normal subgroups that do not contain the commutator subgroup $C$? One can show that any subgroup $H$ that contains $C$ is a normal subgroup with a few algebraic manipulations....
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25 views

Does there exist some sort of classification of finite marginally simple groups?

Let’s call a group marginally simple if it does not have any non-trivial marginal subgroup (strict definition of marginal subgroups and brief overview of their properties can be found here: https://...
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22 views

Characterize edge-transitive Cayley Graphs

Let $G$ be a finite group and $S \subseteq G$ a symmetric subset. The Cayley graph $\Gamma(G,S)$ is always vertex-transitive, but it sufficient a simple example to show that it is not always edge-...
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1answer
21 views

Proof that if $G$ permutes the factors of $T^k$ transitively, $G$ is maximal in $T^k \rtimes G$

Suppose $T$ is a finite non-abelian simple group, $Inn(T^k) \leq G \leq Aut(T^k)$, and $G$ permutes the factors of $T^k$ transitively. Show that $G$ is a maximal subgroup in $T^k \rtimes G$, (where $G$...
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1answer
34 views

Is it possible that a quotient group of G is isomorphic with a subgroup? [on hold]

Let $G$ be a group with $H$ as its normal subgroup. Is it possible that there exists a subgroup of $G$(say $K$) that is isomorphic with $G/H$ ? I believe that $H \subseteq K$, but how it works?
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0answers
19 views

Algorithm to find representatives of orbits when group of permutations acts on set of mappings

Let $N$ be $\{1,\ldots,n\}$, $R=\{1,\ldots,r\}$. Consider mappings from $N$ to $R$. It is clear that there are $r^n$ such mappings. Let $G_n$ be the group generated by $n$ length cycle permutation $\...
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1answer
15 views

The fundamental groups of 3-dimensional spherical space forms

Let $S^3/\Gamma_i\,(i=1,2)$ be a $3$-dimensional spherical space form, where $\Gamma_i \subset SO(4)$ is a finite subgroup acting freely on $S^3$. If $S^3/\Gamma_1$ is homotopy equivalent to $S^3/\...
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1answer
46 views

Why is $\mathbb{Z}$ not an inital object of Gr or AB?

Why is $\mathbb{Z}$ not an inital object of GR or AB? Claim 1: for every group $G$ there exists a groups morphism from $\mathbb{Z}$ to $G$. PF: Let $f:\mathbb{Z} \rightarrow G$ be given by: $f(n) = ...
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1answer
28 views

Are two isomorphic finite subgroups of $SO(4)$ conjugate?

Let $A,B$ be two finite subgroups of $SO(4)$ such that $A$ and $B$ are isomorphic as abstract groups. Can we find a $g \in SO(4)$ such that $$ gAg^{-1}=B? $$ If it is the case, does the same ...
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81 views

How many elements in the cyclic group $C_{54}$ have order $6$? How many have order $8$?

I'm new to the algebra game and I'm learning about group theory at Uni. I understand a cyclic group is a group that can be generated by a single element, but I'm not sure how I would answer the above ...
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1answer
25 views

Prove that $(G,\odot)$ is a group $\iff L$ is a subgroup of $\mathrm{S}_G$

This is Exercise 11 from Section 7: Groups and Homomorphisms, Chapter 1: Foundation, textbook Analysis I by Herbert Amann and Joachim Escher. The proof of $\Longrightarrow$ is quite easy. Please ...
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1answer
24 views

Kernel of ${\rm GL}(n,F)$ on ${\rm PG}(n-1,F)$ over a division ring $F$

I am reading Peter Cameron's note on Classical Groups and I got confused with Proposition 2.1 on page 14. I have no problem in proving that the elements in kernel are scalars. However, I don't ...
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0answers
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Dixon&Mortimer exercise involving size of orbits under subgroup of point stabilizer of given primitive permutation group

I come forth once again with a Dixon&Mortimer exercise (1.5.22), formulated as follows: let $G$ act faithfully and primitively (transitivity implicitly assumed) on finite set $A$ with $|A| \...
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1answer
41 views

Third isomorphism theorem on groups

From this wikipedia's page, I have found these important theorems. I would like to give it a try. Since I am self-learning Mathematical Analysis without teacher or tutor, it will be great if someone ...
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1answer
26 views

Proof explanation of $P(p^e) \cong \mathbb{Z}/(p-1)\mathbb{Z} \times \mathbb{Z}/(p^{e-1})\mathbb{Z}$.

The following is from Classical Theory of Algebraic Numbers by Paulo Ribenboim : $P(p^e)$ is the set of all nonzero residue classes a modulo m, where gcd(a, m) = 1. My question underlined and in ...
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1answer
40 views

spinor representation

I am trying to study Special unitary group of order 2 and some textbooks mention objects transform under special unitary group are called Spinors then How can we represent a spinor using matrix ?
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Finding a normal subgroup 5 (abelian subgroup) [duplicate]

Let (G, ×) be a group, a ∈ G, C5 = {x | xa = ax, x ∈ G}, prove that (C5, ×) is a normal subgroup of (G, ×)
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2answers
69 views

Why does $a^{m_1}=a^{m_2}$ implies $a^{m_1-m_2}=e$?

I was reading this answer. I understand almost all of it. However, there is still one thing that continues to puzzle me. How should I prove for sure that, in this example, if $m_1\neq m_2$ and $a^{m_1}...
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2answers
54 views

Show that $m\mathbb{Z}$ is a subgroup of $n\mathbb{Z} \iff m|n $

Show that $m\mathbb{Z}$ is a subgroup of $n\mathbb{Z} \iff n|m $ I think my solution for one way of this is correct: $\Rightarrow$ Suppose $m \mathbb{Z}$ is a subgroup of $n\mathbb{Z}$ , then $m \...
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0answers
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Order of a spinor [on hold]

Objects transform under SU(2) transformation are called spinors , SU(2) is a ( 2 by 2 ) matrix so order of spinor is ( 2 by 2 ) or (2 by 1 ) ?
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1answer
22 views

Isomorphism between $GL_2(\mathbb{F}_2)$ and $S_3$ [duplicate]

The question is to show that $GL_2(\mathbb{F}_2)$ and $S_3$ are isomorphic where $S_3$ is the symmetric group of $\{1, 2, 3\}$ and the group operation is composition. I have listed all the elements ...
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0answers
43 views

$G$ non-centerless group. How is $Z(\operatorname{Aut}(G))$ made?

Let $G$ be a (possibly infinite) non-centerless group, i.e. such that $Z(G) \ne \lbrace e \rbrace$. Left and right multiplications establish the subgroups $\Theta:=\lbrace \theta_a \mid a \in G \...
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1answer
42 views

isomorphism regarding $\mathbb{Z}$ $\mathbb{Q}$ [on hold]

I know that $\frac{\mathbb{R}} {\mathbb{Z}}$ is isomorphic to the set of all groups of all complex numbers of absolute value $1$ under multiplication Now my question is that $1)$ $\frac{\...
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1answer
22 views

Is there a sum-like operator over subsets of 1 or 2 elements over $\mathbb{Z}$

Let be $A(\mathbb{Z})$ the set of subsets of 1 or 2 elements, like: $\{ 1 \}, \{ 1, 2 \}$. I would like to know if we could prove there is no map $+ : A(\mathbb{Z}) \times A(\mathbb{Z}) \to A(\mathbb{...
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1answer
84 views

Problem 2B.4 in Finite Group Theory by Issacs

Suppose a finite group $G$ has more than one Sylow-2 subgroup and any two intersects trivially. Show $G$ contains exactly one conjugacy class of involutions. Here are some of my thoughts: Suppose not,...
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1answer
32 views

What does addition contribute to the concept of unique factorisation in rings?

A unique factorisation domain is an integral domain $R$ which has unique factorisation. That is, every $x\in R$ has a unique decomposition into the form $\prod p_i$ where the $p_i$ are prime, up to ...
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0answers
15 views

Internal Semidirect with Factors Isomorphic to “Outside” Groups

Here is a conjecture of mine: If $G$ is the internal semidirect product of $N \unlhd G$ and $Q \le G$, and $\phi_1 : N' \to N$ and $\phi_2 : Q' \to Q$ are isomorphisms, then there is some $\theta :...
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1answer
40 views

Presentation of a group generated by reflections through hyperplane

Question: Let $P_i\in\mathbb{R}^n$ be the hyperplane $x_i - x_{i+1} = 0$. Find a presentation for the group $G$ generated by the reflections in $P_1, \ldots, P_{n-1}$ Attempt: I really don't know how ...
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4answers
35 views

Identity element of a group, help [on hold]

So far I have done the RHS to $(x-3)(y-3)+12$ and I have done this to find the identity element, $(e-3)(x-3)+12=x$, then re-arranged to $(e-3)=(x-12)/(x-3)$, now I am stuck.