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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Is there any neat way to show $\phi$ is a homomorphism?

In Michael Artin's Algebra (chapter 2, page 50, example 2.5.13) the author illustrates a homomorphism from $S_4$ (all permutations of indices $(1,2,3,4)$) to $S_3$ (all permutations of indices ...
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0answers
40 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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1answer
30 views

How do generators act on symmetric and antisymmetric representations?

The Lie algebra of a group acts on itself through the commutator $ T_a \in ad$: $$ T_a \circ T_b = [T_a,T_b] \in ad $$ I assume the same should be true if we have an antisymmetric adjoint, as for ...
1
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0answers
23 views

Why is a weight automatically a complex weight?

In the appendix A page 20 of this paper the authors write ...complex weight [ $| (1 / 2 , \sqrt{3/2}) \rangle$], which cannot be accomplished with a single hermitian matrix What do the ...
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0answers
20 views

Why aren't all elements of the $45_a$ representation of $SO(10)$ zero?

We can write elements of the $45_a$, where $a$ denotes antisymmetric, as $10 \times 10 $ matrices, because $$ 10 \otimes 10 = 1_s \oplus 54_s \oplus 45_a$$ Here $10$ denotes the fundamental ...
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1answer
297 views

Characterize normal subgroups - Find all subgroups of $S_3$ conjugate to $\{id, (1,3) \}$ - Fraleigh p. 143 14.29

(27.) A subgroup H is conjugate to a subgroup K of a group G (viz. p. 141 $K \le G$ is a conjugate subgroup of $H$), if $i_g[H] = gHg^{-1} =K$ for some $g \in G$. Show that conjugacy is an ...
2
votes
1answer
88 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 ...
2
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0answers
43 views

$S_6$ contains two subgroups that are isomorphic to $S_5$ but are not conjugate to each other

This is a problem from Ph.D. Qualifying Exams. Show that the symmetric group $S_6$ contains two subgroups that are isomorphic to $S_5$ but are not conjugate to each other. Here is my method. $S_5$ ...
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0answers
14 views

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 $T(D)=\{A\in\mathbb{R}^{(d+1)\times (d+1)} : A\in \text{SO}(d+1), A(X)=X\}$. ...
4
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2answers
75 views

Subgroups of smallest possible index in a solvable group

The following question appears in Isaacs' Finite Group Theory: 3B.15) (Berkovich) Let $G$ be solvable, and let $H<G$ be a proper subgroup having the smallest possible index in $G$. Show that ...
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0answers
18 views

“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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1answer
50 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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1answer
40 views

General form of a matrix $M$ commutes with the unitary representation $U^{\otimes m},~ \forall U\in U(n)$

My question is about the general form of a $n^m\times n^m$ positive definite matrix $M$ where $$[M,U^{\otimes m}]=0,~ \forall U\in U(n)$$ or in other words, M commutes with all members of the the ...
14
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3answers
449 views

Groups having at most one subgroup of any given finite index

Cyclic groups have at most one subgroup of any given finite index. Can we describe the class of all groups having such property? Thank you!
1
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2answers
64 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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1answer
54 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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1answer
44 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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0answers
27 views
+50

Hypercenter is the intersection of normalizers of Sylow subgroups.

I'm trying to prove that the intersection of the normalizers of the Sylow subgroups of a [finite] group $G$ is equal to its hypercenter, i.e., $$Z_\infty(G)=\bigcap\limits_{S\in ...
8
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1answer
55 views

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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0answers
14 views

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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0answers
18 views

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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0answers
34 views

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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3answers
40 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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0answers
6 views

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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4answers
49 views

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: ...
4
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3answers
85 views

Homomorphism from $\mathbb{Z}\oplus \mathbb{Z}\oplus \mathbb{Z}\rightarrow \mathbb{Z}\oplus \mathbb{Z}$ has non-trivial kernel: elementary argument

One can give an elementary arguments (avoiding "rank") to prove that any group homomorphism $f$ from $\mathbb{Z}\oplus \mathbb{Z}$ to $\mathbb{Z}$ has non-trivial kernel: Let $f:(1,0)\mapsto a$ and ...
2
votes
1answer
59 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 ...
1
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1answer
30 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 ...
12
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2answers
203 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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2answers
44 views

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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1answer
45 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 ...
3
votes
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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2answers
45 views

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 ...
0
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0answers
27 views

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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0answers
73 views

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 ...
1
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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 $D$, ...
2
votes
1answer
54 views

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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0answers
53 views

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: ...
3
votes
1answer
49 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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0answers
26 views

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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0answers
62 views

Any hint on : Every $A_{n}$ elemnt is $n$-cycles product.

[Added explanation] I found this exercise as follows in Hungerford : Abstract algebra (3rd edition) page 236, exercise number 40. Stated as follows : C.40. Prove that every element of $A_{n}$ is ...
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0answers
62 views

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 ...
2
votes
3answers
995 views

Why should a group of order 33 have an element of order 3

How do I show that $|G|=33$ has an element of order $3$. I could use Cauchy's and say that since $3$ is prime divisor of $33$ there must be a subgroup of order 3. But this problem shows up in ...
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0answers
90 views

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 ...
0
votes
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 ...
2
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1answer
34 views

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 ...
2
votes
2answers
32 views

Homomorphism with intersection of all Sylow p-subgroups as kernel?

Does anyone know of a homomorphism from a group $G$ to another group with kernel as the intersection of all Sylow $p$-subgroups? I was trying to prove that the intersection of Sylow subgroups is ...
-1
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0answers
37 views

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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0answers
35 views

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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2answers
926 views

Can you always find a surjective endomorphism of groups such that it is not injective?

If we take the following endomorphism, $\phi:R[t] \to R[t]$ by $\sum_{i = 0}^n a_it^i \mapsto \sum_{i = 0}^{\lfloor n/2 \rfloor} a_{2i} t^i$, it is surjective but not injective. (It just removes odd ...