The study of symmetry: groups, subgroups, homomorphisms, group actions.

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“Semidirect product” of graphs?

The first subquestion is "has a standard notion of semidirect product been defined in graph theory"? If yes, i'd like to know if the definition i'm gonna give is equivalent to the standard one. I'd ...
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83 views

Primitive permutation group with subdegree 4

What are the possible sizes of a point stabilizer in a primitive permutation group, where the point stabilizer has an orbit of size 4? In the tradition of subdegree 3 and subdegree 2, I wonder ...
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158 views

Is $H^2$ weakly closed in $H$? Isaacs Finite Group Theory, exercise 8B.6

I'm trying to solve exercise 8B.6 on page 249 of Isaacs's Finite Group Theory textbook (the second question in a series; this is the third as question here). I have an idea, but it doesn't quite ...
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When does PSL(5,q) have order exactly divisible by a specific odd power of 5?

In a misguided attempt at answering a question on divisibility of simple group orders, I looked at $\newcommand{\PSL}{\operatorname{PSL}}v_p(|\PSL(p,q)|)$ which went smooth enough for $p=2$ and $p=3$, ...
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433 views

What does a Cayley table tell?

Other than showing if a group is abelian and the order of the group, what else does a Cayley table indicate?
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176 views

Reference: Geometric group theory

H. Bass has studied existence of lattices on trees. Can someone suggest a (readable) reference for lattices on graphs?
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How do I find the orbits given the ring of invariants?

Suppose I have a group $G$ acting linearly on a vector space $V$ and I know the ring of invariants $K[V]^{G}$ (i.e., I know the subring of $K[V]$ which is fixed pointwise under the induced action). My ...
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302 views

Can “anti-groups” and “anti-manifolds” be constructed? (and other “anti-objects”)

Is it possible to create an "antigroup"? What I mean by this is, given some group G, and some "antigroup" H, then the "free product" of G and H, G*H will equal the "group" (vacuously a group) of no ...
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365 views

A finite length module is the direct sum of the image and kernel of a projection-like endomorphism

I have a question about abelian groups (or rather $X$-groups): Suppose $A$ is an abelian group (or rather an $X$-group) which is artinian and noetherian. $f$ is an endomorphism (or rather an ...
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21 views

Nontrivial relations in rotation groups

Consider the subgroup $H$ of $SO(3)$ generated by rotations of order $5$ (i.e., rotations by $\frac{2\pi}5$) about the $x$ and $y$ axes. This group certainly isn't finite or discrete (as it's not ...
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If $p>2$ is a prime number, Prove that $U(p^k) $ is cyclic

(i) If $p>2$ is a prime number, Prove that $U(p^k) $ is cyclic (ii) Prove that $U(p^n) \thickapprox Z_{(p^n -p^{n-1}) }$ Solution : If I prove that if $U(p^k) $ is cyclic, then, we know that ...
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29 views

~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~

I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ...
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38 views

Existence of a natural transformation

Let $G$ be a group, $S$ a set and $\chi$ an action of $G$ on $S$. This defines a functor $F$ from the group as a single object category to the category of sets. Let $\phi$ be an automorphism of $G$, ...
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50 views

Given a number of items, how many sets of three are there where no two sets are two thirds similar?

Sorry if the title isn't proper math-talk. Hopefully I can explain it better here. So let's say we have a set. 1, 2, 3, 4, 5, 6, 7, 8, 9. I want to know how many groups of three can be made where no ...
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39 views

Isomorphism Classes

I'm trying to find what the isomorphism class of the group of $G = n\times n$ matrices with $\pm 1$ along the diagonal and zeroes everywhere else. My approach was first to show that because for each ...
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36 views

Notation of quotient groups

So I'm determining the quotient group of $(E,+)$ in $(Z,+)$ where E is even int. and Z is int. I know sort of what is happening, I split the group $Z$ into evens and odds (2 sets) and as we are under ...
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50 views

$D_6$, regular hexagon.

Find a subgroup of $D_6$ where $D_6$ is the regular hexagon, with 12 symmetries. $$ D_6 = \{ e, r, r^2, r^3, r^4, r^5, a, ar, ar^2, ar^3, ar^4, ar^5\} $$ where $r^6 = e$. And the $r^n$ represent ...
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64 views

Classification of triply transitive finite groups

A permutation group $G$ on a set $X$ is said to be $k$-transitive if it is both transitive on $X$ and either $k=1$ or the point stabilizer $G_x$ is $(k-1)$-transitive on $X\setminus\{x\}$. Is ...
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26 views

GL$_2(\mathbb{Q}) Z_{\mathbb{R}}$ closed in GL$_2(\mathbb{A})$?

I am struggling with the following subgroups of GL$_2(\mathbb{A})$ where $\mathbb{A}$ is (the topological ring of) Adeles over $\mathbb{Q}$: $$G_\mathbb{Q} := \iota(\text{GL}_2(\mathbb{Q})) $$ where ...
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Center/Commutator Subgroup of Direct Product = Direct Product of these Subgroups - - Fraleigh p. 64 Theorem 6.14

(1.) What's the intuition? Full proof for Center Subgroups (2.) What's the proof blueprint? I know proof's using $A = B \iff A \subseteq B \wedge B \subseteq A$. But where did $(ga,hb)$ in ...
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In a finite cyclic group of order n, number of solutions to $x^m = e$ - Fraleigh p. 68 6.53,54

(53.) Show that in a finite cyclic group G of order n, written multiplicatively, the equation $x^m = e$ has exactly m solutions $x$ in G for each $m \in \mathbb{N}$ that divides n. (54.) With ...
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46 views

$G \simeq R^{\times}$

What is known about the groups G for wich there exist a unitary ring R, such that $R^{\times} \simeq G$? I can easily prove that The only G cyclic with this property(Edit:and odd order) are those who ...
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Commutator Identities in Groups

Let $x$, $y$, $z$ be elements of a group $G$ and let $[x,y]=x^{-1}y^{-1}xy$ be the commutator of $x$ and $y$. Then we have the following identities: $[x,zy]=[x,y][x,z][[x,z],y]$ ...
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66 views

On automorphisms of finite abelian group

Let $G$ be a finite abelian group such that $a, b\in G$ and $\mid a\mid=\mid b\mid$. Then does there exist an automorphism of $G$ such that $\alpha(a)=(b)$? Thank you
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Would the transformation of a differential equation obey the same algebra?

I've found that the algebra of this differential equation $$\frac{d^2y}{dz^2}-(3z^2+\gamma)\frac{dy}{dz}+(cz+\alpha)y=0$$ is in $sl(2)$ because it is possible to use the generators of the $sl(2)$ ...
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40 views

Visualize cosets of kernel of homomorphism, normal subgroup

Question 1. 'Since we know that the codomain is a group, this cannot happen.' I don't understand. Can someone elaborate? I know all homomorphisms are functions but not vice versa. Functions are ...
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27 views

Visualize every quotient map follows a pattern, subgroup and its left cosets

page 167. Because of the Fundamental Homomorphism Theorem, Nathan Carter calls non-embedding homomorphisms quotient maps. This is one of the key facts about homomorphisms: they come in ...
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The group of pure/hereditary sets

From wikipedia: In set theory, a hereditary set (or pure set) is a set all of whose elements are hereditary sets. That is, all elements of the set are themselves sets, as are all elements of ...
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51 views

Clarifying Semi-Direct Products: Example

I'm working through some questions on semi-direct products, and although I can work out these problems (for the most part), I usually have trouble completing them. I have identified some of the things ...
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30 views

Closed subgroups of algebraic group have DCC?

Are there any infinite descending chains of closed subgroups of the general linear group over a field? More specifically, is my argument ok? Can you fill in some of the details? Prop: No. Proof: If ...
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Application of Sylow Theorems.

I have a group theory exam coming up so I was reviewing through my textbook and I stumbled across a problem that looked interesting: Let $p < q$ be two primes and $G$ be a group of order $pq^3$. ...
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32 views

Correspondence between unipotent and nilpotent elements

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. Let $\mathcal{U}(G)$ be the closed subvariety of unipotent elements of $G$, i.e., all elements whose ...
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Semisimplicity of restriction: Representation theory

Let $G$ be a group, $E$ be a vector space over field $K$ and $\rho : G \rightarrow \operatorname{GL}(E)$ a semisimple $K$-representation of $G$. Let $H \lhd G$ be a finite-index normal subgroup of ...
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Reflections in Dihedral Group

In Dihedral Groups, what is the meaning of reflection ? A line needs to be specified for a reflection to take place, but, if you specify only one line how will $D_n$ give all the symmetries for a ...
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Computing the fundamental groups of simple algebraic groups of type $A$

I'm interested in seeing the computation for the fundamental groups of the simple algebraic groups of type $A$. Below is the definition of the fundamental group for a simple algebraic group $G$. Let ...
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An example where $Z(Z_G(A))$ is not a subset of right Engel elements in a finite $p$-group

Find a counter example to the following statement: Let $G$ be a finite $p$-group such that $G/Z(G)$ has exponent $p$. Let $A$ be a normal abelian maximal subgroup of $G$, and $Z_G(A)$ be the ...
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63 views

Showing $|K : H \cap K| \le |G:H|$ where $H,K \le G$

I'd appreciate input on the validity or lack thereof of my attempted proof of the following: Let $H,K \le G$ be subgroups, where $|G:H| < \infty$. a) Show that $|K:H \cap K| \le |G:H|$. b) Show ...
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Link between representation theory and Galois theory: Trivial representation in field towers.

Let $K|F$ be a finite cyclic Galois extension of number fields of degree prime to $p$ with Galois group $H$, where $p$ denotes a rational prime. Let $L|K$ denote a pro-$p$-extension (possibly ...
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Simplifying Relations in a Group

Let $K$ be the group generated by four elements $x_1,\cdots,x_4$ with relations that any simple commutator with repeated generator is trivial; for example, $[[x_2,[x_1,x_3]],x_3]=1$. As I have asked ...
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How “bad” can presentation of the trivial group get?

These questions are sort of preliminary questions and reference requests for a project I am doing. Lets say, for concreteness, that $R$ is a set of words in the free group of rank two and that ...
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67 views

Find the fundamental group and the Alexander polynomial

I would like to find the Alexander polynomial of the link $L$, described below. Let $K(q,r)$ be the $(q,r)$-torus knot embedded on a torus $V$. Inside the torus $V$, consider a smaller solid torus ...
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Polyomino group structure?

Has anyone ever heard of (can you think of) of a group structure on $\mathbb{P}$, the set of all polyominoes? Ideally the monomino would be our identity element, I'd say. (Thanks to Justin Lanier ...
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why does table look like this? - multiplicative table for Group

I have a Group $\mathbb{Z_2}[x]/q$ where $q(x)=x^3+x+1$ In my textbook, there a multiplication table: but I dont understand how the last $x+1$ comes. it should be $(x^2+x+1)*(x^2+x+1) \quad mod ...
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A dense subgroup with completion not isomorphic to the big (pro-p) group?

This is an (early) exercise from the book "Analytic Pro-p groups": (p.31, ex. 3(iii)) Give an example of a finitely generated pro-$p$ group $G$ and a dense subgroup $H$ of $G$, with $H$ finitely ...
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Finitely generated group has only finitely many subgroups of given index

In my previous question i have got a comment that : If a group is finitely generated, then there are finitely many subgroups of a given finite index. I do not yet see the beauty of this problem ...
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$N\unlhd G$ such that $|N|<\infty$ then we have $M\unlhd G$ with $|G/M|<\infty$ and $mn=nm ~\forall m\in M,n\in N$

Now, the question is : Assuming $N\unlhd G$ such that $|N|<\infty$ Question is to prove that : we have $M\unlhd G$ with $|G/M|<\infty$ and $mn=nm ~\forall m\in M,n\in N$ As i am assuming $G$ ...
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Free groups and their commutator subgroups.

I was trying to understand the example 3 of this paper of Baer. Take a free group $F$ of rank 2. Take the commutator group $F'$ (this is a free group having infinite countable rank). Then say $N$ the ...
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Converse of sylow theorem

I have been recently I asked to construct non isomorphic groups of specific order. To do that I used the invariant that they must have isomorphic centers. $G_1\cong G_2 \Rightarrow Z(G_1) \cong ...
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72 views

Subgroups of Cyclic $p$-Groups

Theorem Let $G$ be a cyclic group of order $n$, generated by $x$. Then every subgroup of $G$ is cyclic. The proof is as follows: let $H$ be a proper non-trivial subgroup. Then there exists a ...
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Topology of a 3D wired Mandala?

There is a so called 3D-wired Mandala, based upon $2$ large circles each flowered symmetrically on its circumference by two sets of each $8$ half-circles. The circles are interconnected together by ...