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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Maximally symmetric discrete subsets of the sphere

Let $d\geq 1$ and $\mathbb{S}^{d}=\{x\in\mathbb{R}^{d+1}:|x|=1\}$. For $X\subset\mathbb{S}^{d}$, $|X|<\infty$, let $T(D)=\{A\in\mathbb{R}^{(d+1)\times (d+1)} : A\in \text{SO}(d+1), A(X)=X\}$. ...
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“Super-complement” of a normal subgroup

$\DeclareMathOperator{\Aut}{Aut}$Consider a group $G$ with a normal subgroup $N\triangleleft G$ and suppose that $G$ has a subgroup $H$ so that $G = HN$. Then $H$ acts on $N$; in other words, we have ...
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Is an infinite system of (linear) equations solvable if all finite subsystems are?

I was wondering about the following. Let $A$ be an abelian group, $a_i$ variables indexed with some arbitrary set $I$ and assume we have an infinite set $E$ of linear equations in finitely many ...
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Kernel of homomorphisms of the Baumslag-Solitar group BS(n,-n)

I would like to find the kernel of the following homomorphisms and show that those kernels have trivial intersection. $H:=\mathbb{Z}*\mathbb{Z}/n\mathbb{Z}=\langle p,q| q^n=1\rangle$ ...
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45 views

Is this group of matrices a $p$-group?

Let $R$ be a discrete valuation ring with the maximal ideal $\mathfrak{m}=(\pi)$ and residue field $k$ of positive characteristic $p$. Now consider $\mathrm{M}_n(\pi^iR/\pi^{i+1}R)$, $n\times n$ ...
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28 views

Transformation law for symmetric rank-2 tensors?

A rank-2 tensor $M_{ij}$ transforms as $M_{ij} \rightarrow O_{ik} O_{jl} M_{kl}$, where $O$ is some element of $SO(n)$. We can always get a symmetric tensor from $M_{ij}$ through $M_{ij}^s =M_{ij} + ...
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Quadratic Casimir of SO(5)

In the article A Four Dimensional Generalization of the Quantum Hall Effect, arXiv:cond-mat/0110572, by Zhang and Hu Quadratic Casimir operator for $SO(5)$ is given as $$p^2/2+q^2/2+2p+q .$$ When ...
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About some quotient of finitely generated group

Let $G$ be a finitely generated group. Assume there exists $N$ an abelian normal subgroup of $G$, such that $G/N=H$ with $H$ a finite group of order not divisible by a fixed prime number $p$. In ...
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Is the basis for $\mathbb{R}^n$ where the $\mathfrak{so}(n)$ Cartan elements are diagonal, necessarily complex?

This is a follow-up to this and this question. The elements of $\mathfrak{so}(n)$ are antisymmetric in the standard basis $(1, 0, 0), (0, 1, 0), (0, 0, 1)$. This means that we have no diagonal ...
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2answers
62 views

The relationship between subfields and subgroups of a finite field.

I am trying to get my head around the structure $GF(p^n)$ when viewed as a vector space of dimension $n$ over $GF(p)$ (mainly the relationship between the additive and multiplicative structures). I'm ...
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3answers
39 views

order and cycles of perfect shuffle of 52 cards

This is the shuffle: $$1,2,\cdots,52$$ is turned into $$1,27,2,28,\cdots,26,52$$ when I try to write the cycles of this shuffle, I get a LOT of cycles, for example: $$(2\ 3)(27 \ 2)(26\ ...
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4answers
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How to show that the cycle $(2 5) = (2 3) (3 4) (4 5) (4 3) (3 2)$

I generally do not have any problem multiplying cycles, but I've seen on Wikipedia that $$(2 5) = (2 3) (3 4) (4 5) (4 3) (3 2). $$ I started following the path of $2$ on the right: ...
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1answer
29 views

Partition of a group such that an operation can be defined

I'm struggling with Problem 43 of 3.1 of Dummit's algebra book. The problem is: Assume $P=\{A_i\}$ is any partition of $G$ with the property that a "quotient operation" is defined as follows: to ...
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58 views

Prove that for a group with even order $2k$, there is a subgroup $K$ with order $k$

I'm trying to understand the proof my teacher did: Consider a subgroup $H$ of $G$. If $H$ is not contained in $A_n$, then we can say that there exists at least one permutation in $H$ that is odd ...
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44 views

How many combinations in 10x10x10 Rubik's cube?

I was wondering how many possible combinations there is in the cubes greater than 3x3x3 (4x4x4, 5x5x5, ..., 10x10x10)? We know that in 3x3x3 there are about 4,3 * 10^19 combinations, what about bigger ...
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How to solve conjugation equations in group theory

Given the permutations $(12)(34)$ and $(56)(13)$ find $a$ such that $$a^{-1}xa = y$$ I just realized that I don't know how to solve this exercise. My book don't even give examples of how to solve ...
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Trouble understanding Latin squares and group theory

This is more of a theoretical question, but I'm having trouble understanding why it is that Latin squares are generalizations of a group? I kind of arrived at this question trying to figure out why ...
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3answers
77 views

What does it means to multiply a permutation by a cycle? $\pi(x_1\cdots x_n)\pi^{-1}=(\pi(x_1)\cdots\pi(x_n))$

I have to prove that $$\pi(x_1\cdots x_n)\pi^{-1}=(\pi(x_1)\cdots\pi(x_n))$$ but I can't understand what this means. My book doesn't defines what a permutation and cycle product would be. So, for ...
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1answer
42 views

Finding Automorphisms of Irregular graph through Regular Sub-Graphs.

Objective : To find a set of permutations for a irregular graph which is also a set of automorphism. This finding process uses permutations of 2 regular subgraphs of the given graph. Description and ...
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Some questions about Banach Tarski proof

Banach-Tarski proof as been the topic of a video by the well-known Youtube channel VSauce but there were some parts that I didn't understand. So I went reading for the proof on Wikipedia, and I didn't ...
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A question about semidirect product

When we consider the classification of the group G by semidirect product, we need to consider all the homomorphisms from K to Aut(H), Where G=HK and H$\unlhd$G,H$\bigcap$K=1 But by the theorem: ...
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48 views

Insight about compact groups

I'm quite familiar with the general notion of compactness in math but I have some troubles with its extension to group theory. I'm not talking about definitions or theorems: I would like to have some ...
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1answer
18 views

If $s$ and $t$ are symmetries of a plane such that they agree on three non collinear points then show that $s=t$

This is a problem based on "Symmetry" of the plane $\mathbb{R^2}$. Suppose $A$, $B$, $C$ are the three points in plane which are after the corresponding actions by $s$ and $t$ are in the places $D$, ...
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1answer
52 views

Number of Automorphisms of a Irregular Graph.

I have been looking for results on number of graph automorphisms of irregular graph(upper and lower bound). I searched , but could not find anything which can be used directly. Say, $G$ is $k$ ...
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201 views

Why this $\sigma \pi \sigma^{-1}$ keeps apearing in my group theory book? (cycle decomposition)

I'm studying cycle decomposition in group theory. The exercises on my book keep saying things like: Find a permutation $\sigma$ such that $\sigma (123) \sigma^{-1} = (456)$ Prove that there is no ...
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Prove that there is no permutation $\gamma$ such that $\gamma (1 2) \gamma^{-1} = (1 2 3)$

I need to prove that there is no $\gamma$ such that: $$\gamma (1 2) \gamma^{-1} = (1 2 3)$$ First of all, I'll try to write $\gamma$ in a generic way: $$\gamma = (a b c) \implies \gamma^{-1} = ...
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Symplectic group and Quaternion Inner product

I have a problem understanding a passage from "Naive Lie theory"(by Stillwell), here is the passage from section $3.9$ ,page $71$: The idea of treating orthogonal, unitary, and symplectic groups ...
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1answer
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Direct Limit of finitely generated groups

Is every group the direct limit of its finitely generated subgroups? This is true for abelian groups, I have not seen this statement for nonabelian groups, so i am wondering if this is true. Seems ...
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Show that $Nil(\mathbb{Z}_n) $ is a subgroup of $\mathbb{Z}_n$ [on hold]

Show that $\mathrm{Nil}(\mathbb{Z}_n) = \{\bar{x}\in \mathbb{Z}_n\mid \bar{x}\,^m=\bar{0}\text{ for some positive integer $m$}\}$ is a subgroup of $\mathbb{Z}_n$.
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1answer
28 views

Can two representations with different dimensions be isomorphic?

For a finite group G and two irreducible representations, with different dimensions. How would I show that they can not be isomorphic?
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1answer
72 views

If $g$ is a permutation, then what does $g(12)$ mean?

In Martin Lieback's book 'A Concise Introduction to Pure Mathematics', he posts an exercise(page 177,Q5): Prove that exactly half of the $n!$ permutations in $S_n$ are even. (Hint: Show that ...
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Curtains and groups

This picture is a copy of the pattern on my curtains. The points of a hexagonal lattice are each coloured with one of four possible colours. It has translational symmetry in two directions: a ...
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Proofs of Sylow Theorems [on hold]

‎let ‎‎$‎G‎$‎‏ ‎is a‎‎ ‎finite‎ group and ‎$‎‏p‎$‎ is prime. if ‎$‎P‎\in Sly‎_{p }(G)‎$‎‏‎‎ then ‎$‎O‎_{p}(G)=Core‎_{G}(P)‎$‎‏‎‎
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problem in meaning of symbol in commutator subgroup

i was reading paper "OUTER AUTOMORPHISMS IN NILPOTENT p-GROUPS OF CLASS 2, H. LlEBECK" in page 2 there is a symbol i dont get. if G is generated by a basis $a_λ, λ∈Λ$ and z∈Z⋂Φ(G) for σ(z,μ) be ...
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Permutations minus Transpositions

I want a formula that allows me to find all the permutations in $S_n$ (which is the set of all the integers from 1 to $n$) which don't contain a transposition. Attempt: Lets call $g(n)$ the ...
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1answer
84 views

What finitely generated amenable groups are known to be LERF?

I know that finitely generated nilpotent groups are LERF (LERF means "subgroup separated"). I'm looking for examples (many, if possible) of groups which are: Finitely generated, but infinite ...
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Why is reflection in a plane an automorphism?

I have not studied group theory, but would like to know in simple terms why reflection in a plane is an automorphism. Dr. Hermann Weyl gives the definition of automorphism in his book 'symmetry' as ...
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How is the distinction of left and right in space related to the orientation of screw?

In Dr. Hermann Weyl's book 'symmetry', he explains the difference between left and right as In space the distinction of left and right concerns the orientation of a screw. If you speak of turning ...
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1answer
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Bijection from conjugacy class to the factor group by centralizer.

How different is $g^{-1}xg$ from $gxg^{-1}?$ Because proving a bijection from $g^{-1}xg$ type conjugacy class to the set of right cosets of the centralizer of $g$ in $G$ is as easy as proving it from ...
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Composition series and its number determine a group?

By Jordan-Holder thm, it is known that every finite group has a unique composition series.(Here, unique means that there is only one kinds of such series.) And it is known also that composition ...
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Zero-Sum Partitions of Nonzero Elements of a Ring

In this question, rings are not necessarily finite nor do they need to be unital (i.e., the multiplicative identity may not exist). Although I shall almost exclusively discuss finite commutative ...
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Ideals of non semi-simple group rings.

I worked for a long time on complex group rings and complex twisted group rings. In those cases the algebra is semi-simple and its structure is well understood from the decomposition to irreducible ...
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Given the basis vectors of a 10-dimensional representation of $SO(10)$, how can I compute the basis vectors of the 54-dimensional representation?

Because $10 \otimes 10 = 1_s \oplus 54_s \oplus45_a$ we can write each element of $54$ as a $10×10$ matrix. The usual basis vectors of the 10-dim rep are $$ \begin{pmatrix}1 \\0 \\ \vdots ...
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Simple example for Bilinear mapping

Notation : $\mathbb{G}$ is an additive group and $\mathbb{G}_T$ is multiplicative group of prime order $q$. Bilinear mapping $e: \mathbb{G} \times \mathbb{G} \rightarrow \mathbb{G}_T$ has to satisfy ...
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Elements of $S_n$ which can not be product of $\leq n-2$ transpositions

It is well known that every element of $S_n$ can be written as a product of at most $n-1$ transpositions. This theorem is proved in all the books which discuss the permutation groups. But, I find that ...
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$Ker (f)$ is finite, then $G$ is finite. [on hold]

Let $G$ be a group with identity element $e$, $f: G → G$ a homomorphism for which there is a natural $n> 1$ such that $f^n (G)$ = {e}. i. Prove that if $Ker ...
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Does $G\times H\cong G'\times H'$ imply $G\cong G'$ and $H\cong H'$?

I know that $G\cong G'$ and $H\cong H'$ implies $G\times H\cong G'\times H'$. But is it true for reverse? I mean, does $G\times H\cong G'\times H'$ imply $G\cong G'$ and $H\cong H'$? If so, how to ...
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If a set $X$ contains three different elements $a,b,c$ describe $f:=t(a,b)∘t(b,c)$ and $g:=t(b,c)∘t(a,b)$. Are they equal?

The group of permutations of a set $X$ consists of all functions $f:X\to X$ that are one-to-one and onto. The group operation is the composition of functions. Of special importance are transpositions ...
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Why doesn't the “naive” scalar product for $SO(n)$ yield something invariant?

By definition, for $SO(n)$ we have $g^T g=1$ for $g \in SO(n)$. Given some vector $v \in V$ and some representation $R: SO(N) \rightarrow \mathrm{Lin}(V)$, the defining condition above tells us ...
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Under what conditions is a ZG-module torsion-free?

If we have a ZG-module A, I was wondering if there are known condition we may imply on either A or the group G to make A torsion-free?