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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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20 views

Dihedral and permutation groups isomorphism

Why dihedral group $\mathbb{D_4}$ is isomorphic with $\mathbb{S_4}$ and not isomorphic with $\mathbb{S_8}$?
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1answer
28 views

Is the symmetry group of the square isomorphic to $\mathbb{Z_8}$? to $\mathbb{S_8}$? to a subgroup of $\mathbb{S}_8$?

Exercise: Mark True or False. Explain why. a) The symmetry group of a square is isomorphic to $\mathbb{Z_8}$. b) The symmetry group of a square is isomorphic to $\mathbb{S_8}$. c)The ...
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1answer
16 views

Group action on convex cone

I was wondering if I could get some help understanding the following fact in an academic paper. The setup is as follows: An open subset of $\Omega \subset \mathbb R^k$ is an open convex cone if it ...
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0answers
27 views

Center of Carnot Group [on hold]

I know the center of the Heisenberg group (over $\mathbb{R}$) is isomorphic to $\mathbb{R}$, but what about the wider class of Carnot groups? Are their centers also trivial?
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1answer
34 views

Finding a set on which a group acts on

I have a given group, defined by its table, namely: Now I am asked to find a set $X$ where this group acts upon non trivially. I have problems understanding this question. As I understand, I am ...
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1answer
40 views

Proving $S_3$ is not a coproduct of cyclic groups

Aluffi II.6.11 suggests proving the above. Here's the sketch of my proof by contradiction. Assume that $S_3$ is a coproduct of a family $\mathcal{C}$ of cyclic groups $C^i$. Using the universal ...
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1answer
34 views

Elementary question on quotient group construction

I am going though an introductory proof text book and was hoping if someone could verify/correct my attempt at the following problem : Let $(\Bbb{Z}[\sqrt3],+,.)$ where $\Bbb{Z}[\sqrt3] \subset\Bbb{...
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1answer
43 views

An exercise in an introduction to the theory of groups [on hold]

4.4 Let $G$ be a finite p-group; show that if $H$ is a normal subgroup of G having order $p$, then $H$ is a subgroup of $Z(G)$. $Z(G)=\left\{x\in G| xy=yx, \forall y\in G \right\}.$ Can you help me ...
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1answer
43 views

Group Theory - Natural Coordinate Function

I am studying Shattered Symmetry - From the Periodic Table to the Eightfold Way; in order to start getting into group theory, and quantum mechanics. I am struggling to understand what the author ...
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1answer
64 views

Sum of subsets is the whole set?

Let $A$ be subset of $\mathbb{Z}_n$. Define $A \oplus A = {\{a \oplus a’ : a \in A, a’ \in A}\}$, where $a \oplus a’ = a+a \mod n$. If $A$ is not contained in a coset of a proper additive ...
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0answers
43 views

Quadrants of a cyclically ordered group

I am trying to prove that if there are two different positive elements in a non-linearly cyclically ordered group, then every quadrant of the group is not empty. Is this a correct statement? ...
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1answer
51 views

Doubt regarding “Elementary approach to proving that a group of order 9 is Abelian”

I am trying to understand the solution of this problem . I am unable to understand why : If $yx=x^2y$, then $yxy^{-1}=x^2$. This means that $y^3xy^{-3}=x^8 $ It seems like I am missing something ...
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0answers
23 views

Lattices are not solvable in non-compact semisimple Lie groups

I'm trying to prove the following result. If $G$ is a non compact semisimple Lie group (lying in some $SL(l,\mathbb{R})$), and $\Gamma$ is a lattice in $G$, then $\Gamma$ is not solvable, and $[\...
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1answer
51 views

Why is $\mathbb{Q}[i]$ not a subgroup of $\mathbb{R}$?

$i$ is the imaginary unit on the complex plane. I am confused on a particular portion of the definition of a subgroup. Let $G$ be a group and $H$ is a subset of $G$. A part of the definition is ...
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1answer
49 views

The number of group homomorphisms from $\mathbb{Z_2}\times \mathbb{Z_2}\rightarrow \mathbb{Z}_4$

Find the number of group homomorphisms from $G_1:=\mathbb{Z_2}\times \mathbb{Z_2}\rightarrow G_2:=\mathbb{Z}_4$ Let $a,b$ be the generators of $G_1$. Then $0=f(0)=2f(0)$ so we have $f(a)\in \{0,2\}$ ...
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1answer
31 views

$|G|=p^2q^2$ where p,q are primes, $p<q$ and $p \nshortmid (q^2-1)$ Show $G$ is Abelian.

$|G|=p^2q^2$ where p,q are primes, $p<q$ and $p \nshortmid (q^2-1)$ Show $G$ is Abelian. ($p$ does not divide $q^2-1$) The first thing to note is that $(q^2-1)=(q-1)(q+1)$ so we can conclude that ...
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2answers
28 views

Infinite abelian subgroup of $SL(2,\mathbb{C})$ [on hold]

Are there any infinite abelian subgroups of the special linear group $SL(2,\mathbb{C})$?
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0answers
21 views

What is $\ell^2(\Gamma)$ for a discrete group $\Gamma$?

I am trying to get my head around the left regular representation of a group, and I am not sure of the definition of the space $\ell^2(\Gamma)$ if $\Gamma$ is a discrete group. To quote what I am ...
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1answer
40 views

Nonabelian group of order $p^n$ for every $p$ prime and $n \in \mathbb{N}$

Nonabelian group of order $p^n$ for every $p$ prime and $n \in \mathbb{N}$ My idea was to just create some sort of generalized dihedral group, that's what I called it at least, maybe it is. $G = <...
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0answers
31 views

Is there a name for a group describing universal symmetry? [on hold]

I am looking for the name of a group which is the opposite of the trivial group. The trivial group is a group describing the symmetries of a transformation where only the identity transformation ...
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2answers
36 views

Help with homomorphism

Can someone please explain where the equation on the RHS of ⇒ came from in the first line of the maths solution? Question: A group G is called simple if it is not the trivial group and its only ...
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0answers
34 views

For a group morphism $G \rightarrow H$, give the image of the generators.

I have two group tables, both with order $24$. Now I have to write the images of the first generator as the second generator. The group elements are all letters, thus $G=\{A,B,C,...,V,W,X\}$ and the ...
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1answer
29 views

What is the radical of upper triangular matrices?

Let $B$ be the algebraic group of upper triangular matrices with entires in some algebraically closed field. I would like to know what is the radical of this group is... Any explanation would be ...
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0answers
47 views

What if the sum in the RHS of the Class Equation is extended to the whole $G$?

For a finite group $G$ of order $n$, the Class Equation reads: $$n=\sum_{b_j \in B}[G:C_G(b_j)]$$ where $B$ is a set of representatives of the conjugacy classes of $G$. Q: Can else come from ...
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0answers
41 views

Embedding a polynomial ring into $\mathbb{Z}_{n^s}$

My setting is the following: $n$ is a product of two big primes (RSA-like), and I am given a $R = \mathbb{Z}/n^s\mathbb{Z}$ as a space to work with. I would like to represent elements of $R$ as ...
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3answers
177 views

Prove that every group of order 15 is abelian? [duplicate]

I had seen this proof at many places, but everywhere sylows theorem is used. So is their any way to solve it without using sylows theorem?
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1answer
89 views

Axiomatisable Groups

Let A be a set of sentences (“proper axioms”) in a first-order language L with equality. Let us write Mod(A) for the class of all models of A which respect equality. We say that a class of L-...
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0answers
36 views

A dodecahedron out of five tetrahedra AKA Partitioning an orbit in X/K into orbits under cosets of H in K

Consider the orbit space $X/K$ with $X$ a symmetric space and $K$ a group. Let $x$ represent an orbit $Kx$ in $X/K$. Now let's introduce a subgroup $H \subset K$, split up $K$ into cosets $aH$ (with $...
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0answers
36 views

embedding a proper normal subgroup of a p-group

I am trying to prove this claim: Let $G$ be a finite $p$-group and $H$ be a proper normal subgroup of order $p^k$,then $H$ can be embedded into a normal subgroup of order $p^{k+1}$. Here is my ...
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1answer
36 views

How to prove $Q_8$ is not subgroup of $S_4$ by sylow thm?

Prove $Q_8$ is not subgroup of $S_4$ Hi. It is trivial that $D_4$ is a subgroup of $S_4$. But the case $Q_8$ make me confuse why this group is not a subgroup of the $S_4$. Why is the $Q_8$ isn't ...
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1answer
70 views

$H^1(SO(2),T(2))$, or finding all subgroups of the group of motions of the plane which are complementary to the translation subgroup

My question is motivated by this part of 'Basic Notions of Algebra' from Igor R. Shafarevich. He mentions the group of motions of the plane in the following context. Let denote $SE(2)$ the 2-...
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0answers
140 views

Raising to the power of $i$

We all know the usual exponentiation $a^i$ in the complex setting; one can ask is there such a map in a given group; more specifically is there a criterion in the general sense such that given an ...
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2answers
36 views

Inductive Proof of Group with Prime Decomposition is Isomorphic to Direct Product of Cyclic Groups

My lecturer set as a bonus exercise the following induction proof: If $G$ is a finite abelian group $|G| = p_1^{n_1} \cdots p_s^{n_s}$ is the decomposition of $|G|$ into a product of distinct prime ...
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0answers
22 views

Group structure on the set of maps in a model category

Let $B$ be an object equipped with two maps $ \mu: B \times B \to B$ and $\rho : B \to B$ in a cocomplete model category $\mathcal{M}$ such that $(B, \mu, \rho )$ is a group object. These two maps ...
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0answers
20 views

Generators of the second quotient of the lower central series

Let $F$ be a free group on $S=\{ x_{1},x_{2},\dots,x_{n}\}$ and let $y_{i}=x_{i}F'$ be the coset classes of $x_{i}$, where $F'=[F,F]$ the derived subgroup. Is the set of commutators among $y_{i}$ a ...
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1answer
36 views

Proving every subgroup of a nilpotent group is subnormal

Let $G$ be a nilpotent group, i.e. $\exists$ an upper central series: $1\triangleleft Z_1(G)\triangleleft Z_2(G)\triangleleft ....\triangleleft Z_k(G)=G$ Where $\frac{Z_{i+1}}{G}=Z(\frac{Z_i}{G})$ ...
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1answer
29 views

Composition of elements in dihedral group

I've come across the following example: $$ρ^3·σρ^2 = ρ^2σρ^{−1}ρ^2 = ρ^2σρ = ρσρ^{-1}ρ = ρσ = σρ^{−1} = σρ^5$$ And was wondering if it is true in general that $ρ^i·σρ^j = σρ^{i+j}$? I know that $ρσ =...
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1answer
41 views

Proving group objects in the category of sets are group [on hold]

I'm having difficulty proving that in the category of sets, the group objects are just groups. I know that for a category, C with finite products, an object,G, is a group object with the morphisms m,e,...
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0answers
32 views

X ={Cg | g has order 2}. Find that cardinality of X. [on hold]

$GL_n(F)$ be the group of all invertible matrices over a field $F$. $C_{g}$ is defined as {$hgh^{-1} : h\in GL_n(F)$}. Let $X$={$C_g$ | $g$ has order $2$}. Find the cardinality of $X$.
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1answer
30 views

Isomorphism of a quotient group and a subgroup [on hold]

Let $G_1$ be a finite group with a subgroup $G_2$ and a normal subgroup $H$. Suppose that $G_1/H$ is isomorphic to a subgroup of $G_2$. Do we have $G_1/H$ is isomorphic to $G_2$?
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0answers
25 views

Under what condition is the group G isomorphic to the direct product of irreducible representations in a block diagonal representation? [on hold]

Given an injective completely reducible representation D(g) of G. Using a smiliarity transformation S, $SDS^{-1}$ is in block diagonal form with ireducible representaion $D_1 , D_2, D_3 .... D_n$. ...
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0answers
43 views

Are there ways to combine manifolds that form groups? [on hold]

I'm thinking of group elements that might be labelled by 3-manifolds. So for example if $[m]$ is a group element. Then the group operation would be $[m]\times[n]=[p]$. For some manifolds $m,n,p$. ...
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2answers
114 views

Is this an appropriate application of the Orbit-Stabilizer Theorem?

Let $G$ be a group and $H \unlhd G$. Then, $\forall h \in H$: $$[G:C_G(h)]=|O_h|$$ where $C_G(h)$ is the centralizer of $h$ in $G$ and $O_h$ is the conjugacy orbit by $h$ (Orbit-Stabilizer Theorem). ...
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0answers
49 views

GAP code to calculate the a certain subgroup $E(G)$ of a group

I am a research scholar from India. At present, I am working on a problem. For this problem, I need to construct the subgroup $E(G)$ of a group $G$ in GAP. Please help me. My question is as follows: ...
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0answers
54 views

To prove the $(\Bbb R , +)$ is isomorphic to $(\Bbb C , +)$ without using metric space? [duplicate]

Firstly, I want to show $(\Bbb R , +)$ is isomorphic to $\Bbb R^2$ (under component wise addition). In this I used the mapping $\phi_1$ : $\Bbb R^2 \rightarrow \Bbb R$ Where, $\phi_1(a,b) = 2^a3^...
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1answer
40 views

All minimal subsets of $(X,G)$

Let $X=[0,1]$ and $G$ be the group of homeomorphisms on $X$. I want to find all the minimal subsets of $(X,G)$. Actually, here $(X,G)$ is a transformation group and a minimal set in this set is a set ...
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2answers
25 views

Decide whether the following subsets of a vector space are sub-vector spaces.

a) $\{f\in\mathbb{R}[t]:f(1)=0\}=:U_1$ b) $\{f\in\mathbb{R}[t]:\exists a\in\mathbb{R}\text{ with }f(a)=0\}=:U_2$ where $\mathbb{R}[t]$ is the set of all polynomials above K. Does a) mean, that ...
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0answers
23 views

An isomorphism between $A/P^{e}A$ and $A/PA$ and the direct sum of a $p$-group

I'm making some notes for Number Theory in Function Fields by Rosen, and for the life of me I cant figure out why the following is true (in the notation below $A = \mathbb{F}_{q}[T]$, $P \in A$ is ...
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1answer
46 views

Prove that if $o(a)=m, o(b)=n$ and $gcd(m,n)=1$ then $o(ab)=mn$

Prove that if $o(a)=m, o(b)=n$ and $gcd(m,n)=1$ then $o(ab)=mn$ I am asking this because I saw someone asked a similar question and was told that the statement is wrong. I managed to show quite ...
2
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2answers
67 views

How to tell if an element is in a cyclic group?

Let $G=<a>$ be a cyclic group. $O(a)=24$. We will define $H_k$: $H_k=<a^k>$. Find for what $k$ values, $0\leq k \leq 23$, $a^5H_k=a^{13}H_k$. I couldn't figure a way to answer this ...