Tagged Questions

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 studies groups.

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Automorphism group for a family of bigraphs

Extend the sides of a regular polygon on 2m vertices, m ≥ 2, to define the 2m(m - 1) finite points of intersection. Circumscribe a centrally symmetric circle large enough all of the points of ...
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An example of a group, a subgroup and an element, satisfying a given condition. [duplicate]

Is there a group $G$, a subgroup $H$ and an element $x$, such that $xHx^{-1} \subset H$, but $xHx^{-1} \neq H$? Thanks in advance.
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Visualize $\mathbb{S}^3/\Gamma$!

I thought the only 3-manifold with positive constant curvature is $\mathbb{S}^3$. But I faced $\mathbb{S}^3/\Gamma$, where $\Gamma$ is a finite subgroup of $SO(4)$ and surprised! My problem is that I ...
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What is the smallest $n$ for which the usual “counting sizes of conjugacy classes” proof of simplicity fails for $A_n$?

A common proof of the simplicity of $A_5$ proceeds as follows. First, note that a normal subgroup is always a union of conjugacy classes. Also, a subgroup has order dividing the size of the group by ...
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Subgroups of amalgamated free product

My question is the following: Suppose that we are given the amalgamated product $G = G_1 * _{G_3} G_2$, and subgroups $H_i \le G_i$ for $i=1,2,3$, such that in addition $H_3$ is as large as ...
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multi-dimensional numbers

Questions: I am trying to derive a multi-dimensional number system. I know it is not the traditional way of doing things; my questions are: Is this valid? If not where are my mistakes? Has someone ...
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Transposition $(a\ b)$ with $\gcd(b-a,n) =1$ and cycle $(12\dots n)$ generate $S_n$

Prove that if $1≤a<b≤n$ with $\gcd(b−a,n)=1$, then the transposition $(a\ b)$ and the cycle $(1 2…n)$ generate $S_n$.
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Problem on abelian group

Let $G$ be an abelian group, and $\Phi:G\to \mathbb{R}$ is a function with the following property: $$\forall a,b\in G,~~ |\Phi(a+b)-\Phi(a)-\Phi(b)|<c$$ The problem asks to prove the existence of ...
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How to prove that if $p^k$ divides a finite group $G$ there exists a strict subgroup $p^k||H|$ or $p|Z(G)$

Given a finite group $G$ and a prime $p$ such that $p^k| |G|$. Now prove that there exists a strict subgroup $H$ of $G$ such that $p^k | |H|$ or $p| |Z(G)|$. Well, I know what you're thinking: just ...
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If $H$ and $K$ are conjugate in $G$, are they conjugate in

Suppose $G$ is a group, with $H$ a subgroup. Suppose that $K$ and $L$ are subgroups in $H$ that are conjugate in $G$, so we have an element $g\in G$ with $gKg^{-1}=L$. Does it follow that $K$ and ...
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Geometrical interpretation of a group action of $SU_2$ on $\mathbb S^3$

Background There're some nomenclatures from Michael Artin's Algebra to explain. 3-Sphere, or $\mathbb S^3$, is the locus of $x_0^2+x_1^2+x_2^2+x_3^2=1$, where $(x_0,x_1,x_2,x_3)\in\mathbb R^4$. ...
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Group homomorphisms into groups with addition?

Are there group homomorphisms other than logarithm into groups with addition? What I mean is, suppose $h(x*y) = h(x) + h(y)$ are there other $h$ and $*$ than logarithm and multiplication that ...
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Image of subgroup and Kernel of homomorphism form subgroups

Is my proof ok? Let $f:G\to G^{\prime}$ be a group homomorphism and let $H\lt G$. $Im(H) = \{f(x):x\in H\}$. To show that $Im(H)$ is a group, it suffices to show that $f(x)f(y)^{-1}\in Im(H)$. ...
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How many $p$-elements does $G$ have?

Let $G$ be a group with a non normal $p$-sylow subgroup $P$. Is there any information about the number of $p$-elements(an element with $p$-power order) of $G$?
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Can every set be a group? [duplicate]

Lets consider a set A. Can we prove that there exists at least one operation * such that under it the set forms a group. May I know how to approach this question?
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If $N \unlhd G$, $G/N$ solvable, $\chi \in Irr(G)$, $\psi \in Irr(N)$ and $<\chi_N , \psi>_{_N} \neq 0$, then $(\chi(1)/\psi(1)) | [G:N]$.

If $N \unlhd G$, $G/N$ solvable, $\chi \in Irr(G)$, $\psi \in Irr(N)$ and $\langle\chi_N , \psi\rangle_{_N} \neq 0$, then $(\chi(1)/\psi(1)) | [G:N]$. Can anybody help me to prove that?
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Exponent and abelian groups

What exponent $e$ can guarantee the group to be abelian? Are there any known results except the case $e = 2$? Thanks for help.
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Constructing the character table of a group

I am aware that, given a group, there is no simple general procedure to construct the character table of the group (over complex numbers). However, for specific groups, we could use helpful additional ...
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Computing Brauer characters of a finite group

I am studying character theory from the book "Character Theory of Finite Groups" by Martin Isaac. (I am not too familiar with valuations and algebraic number theory.) In the last chapter on modular ...
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Purpose of cusps

In the theory of of modular forms, there is the set of of cusps defined by $\mathbb{P}^1 (\mathbb{Q})= \mathbb{Q} \cup \{\infty\}$. For an subgroup $\Gamma < \text{SL}_2(\mathbb{Z})$ of finite ...
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Conjugacy Class of symmetry group $S_{10}$

Let $X=\{a\in S_{10} | ~~\text{order}(a)=8\}$. Determine how many conjugacy classes are in $X$. How to do this question in general?
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What is the quotient of a cyclic group of order $n$ by a cyclic subgroup of order $m$?

Suppose that $H$ is a (cyclic) subgroup of order $m$ of a cyclic group $G$ of order $n$. What is $G/H$? It's a very simple question but I am still struggling with getting accustomed to the ...
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What is this “Schreier factor function”?

I'm now reading the paper "Automorphisms of Group Extensions" by C. Wells [Trans. Amer. Math. Soc. 1971]. There is a paragraph mentions about the Schreier factor function $\mu\colon\Pi^2\to G$ ($\Pi$ ...
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Kernel of this action is trivial?

Let $S_n$ act on $S_n/H=\{ H,t_2 H,...,t_e H\}$ from the left and let $n\geq 5$, $e \geq 3$. Then the kernel of this action is trivial. This supposedly follows because $e \geq 3$, but I haven't the ...
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Understanding Basic Categorical Duality with an Example from Group Theory

I am trying to understand the concept of duality in category theory, but I am having a problem, well illustrated by the following situation. Let $H$ be any nontrivial subgroup of the alternating ...
From Advanced Modern Algebra (Rotman): Proposition 4.10 If $G$ is an abelian group and $p$ is prime, then $G/pG$ is a vector space over $\Bbb{F}_p.$ Definition If $p$ is prime and $G$ is a ...