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

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Fixed Spaces for Group Elements

what is the GAP code for finding the fixed space? A list of row vectors that form a base of the vector space $V$ such that $v M = v$ for all $v$ in $V$ and all matrices $M$ in the list $mats$.
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3answers
37 views

orthogonal and special orthogonal group of dimension $2$, group of isometries of $S_1$, $\mathbb{R}^2$ [on hold]

In my abstract algebra class, my teacher gave us this problem as to help review for the final. Unfortunately, I am not very well versed with linear algebra so I don't understand all that well what ...
4
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1answer
51 views

If a nilpotent group $A$ act on a group $G$ then $G$ is solvable.

Theorem: If a nilpotent group $A$ act on $G$ by automorphism and $C_G(A)=e$ then $G$ is solvable. I hope that the statement of theorem is true. It should belong to Hartley, but when I googled it I ...
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1answer
44 views

Compute the index of $\mathbf{Z}[\alpha]$ in $\mathbf{Z}[\alpha,\beta,\gamma]$

How do I compute the index of $\mathbf{Z}[\alpha]$ in $\mathbf{Z}[\alpha,\beta,\gamma]$? For example, $\alpha = \sqrt[3]{-19}$ and $\beta = (\alpha^2 - \alpha + 1)/3$ satisfy $(\alpha + 1)\beta = ...
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1answer
53 views

For $G$ an abelian group and $H$ a subgroup, is $[G : H]$ the smallest positive integer $n$ such that $ng \in H$ for all $g \in G$?

Let $G$ be an abelian group and $H$ a subgroup. What is the smallest positive integer $n$ such that $ng \in H$ for all $g \in G$? Is it $[G : H]$, or can it be strictly smaller (a divisor of $[G : ...
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0answers
35 views

Algebraic Structures Books [duplicate]

I wanted to ask you guys if you know any books where I can learn basic stuff about Algebraic Structures and Groups. Thank you.
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3answers
55 views

Quotient Objects in $\mathsf{Grp}$

I don't know how to precisely formulate my question, but here goes: Subobjects and quotient objects are duals, so a quotient object in $\mathsf{Grp}^{\text{op}}$ is a subobject in $\mathsf{Grp}$. The ...
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2answers
29 views

S3 group action faithful?

I'm struggling with understanding the term "faithful". I read that a group action for example $S_3$ is faithful on {1,2,3}. Does that mean $S_3$ is not faithful on {1,2,3,4} because it never changes ...
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1answer
154 views

What is a Simple Group?

I am working on this problem from class note: Let $G'$ be a group and let $\phi$ be a homomorphism from $G$ to $G'$. Assume that $G$ is simple, that $|G| \neq 2$, and that $G'$ has a normal ...
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1answer
26 views

twists of unipotent algebraic groups

Let $U$ be a unipotent linear algebraic group over some field $k$ with char$k$=0. Let $U'$ be a linear algebraic group over $k$ such that $U'_{\bar{k}} = U_{\bar{k}}$ (ie $U'$ is a $\bar{k}/k$-twist ...
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39 views

a question about abstract algebra, prove that $HK\cong H\times K$

Let $H$ and $K$ be subgroups of a group $G$, $HK=KH$ and $H\cap K=\{1\}$. Prove that $$HK\cong H\times K.$$ Can some one tell me how to prove this question? I have spent too much time in it, ...
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4answers
127 views

To which group is $G/\ker \phi$ isomorphic to?

$G = \langle H, \odot_7\rangle$ where $H= \{ 1,2,3,4,5,6\}$ and $\odot_7$ denotes the operation, multiplication modulo $7$, the function $\phi: G \to G$ defined by $\phi(g)=g^2$ . List the elements ...
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1answer
17 views

Prove that for a general 3-cycle \sigma one can find a permutation …

I need to prove that for a general 3-cycle $\sigma$ one can find a permutation $\tau \in S_4$ such that $\tau \sigma\tau^{-1} = (123)$, and use this to show that 3-cycles in $S_4$ are even. How do i ...
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1answer
33 views

Prove that $\def\Aut{\operatorname{Aut}}\Aut(\mathbf{Z_{n}})\simeq \mathbf{Z_{n}^{*}}$

I am writing another exam in Algebra this week and this time the main topic is automorphism. I was again going through the example exercises and exams from previous years and this problem is giving me ...
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0answers
36 views

$n_p(GL_2(\mathbb{F}_p))=p+1$

I'm interested in the following problem from Dummit & Foote's Abstract Algebra text (Exercise 40 of Section 4.5): Prove that the number of Sylow p-subgroups of $GL_2(\mathbb{F}_p)$ is $p+1$. ...
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26 views

A finitely generated soluble group isomorphic to a proper quotient group

Let $\mathbb{Q}_2$ be the ring of rational numbers of the form $m2^n$ with $m, n \in \mathbb{Z}$ and $N = U(3, \mathbb{Q}_2)$ the group of unitriangular matrixes of dimension $3$ over $\mathbb{Q}_2$. ...
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1answer
28 views

Direct Product of Cyclic Groups and Quotient Groups

Let G = $Z_4$ x $Z_6$ be the direct product of cyclic groups $Z_4$ and $Z_6$. Let N = <(2,3)> be a normal subgroup of G. Show that G/N $\simeq$ $Z_{12}$ What I have so far.. Given that |N| = 2, ...
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2answers
45 views

$S$ be the collection of groups $G$ in which every element in $G$ commutes only with the identity element and itself [on hold]

Let $S$ be the collection of (isomorphism class of ) groups $G$ which have the property that every element in $G$ commutes only with the identity element and itself. Then A. |S|=$1$ ...
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2answers
49 views

Direct Product of Two Rings and 0-divisors

Let $R = Z_4\times Z_3$ be the direct product of $Z_4\times Z_3$. Find all 0-divisors in $R$. So far, I have these elements as the product of $Z_4\times Z_3$ : $(0,0)$, $(0,1)$, $(0,2)$, $(0,3)$, ...
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1answer
46 views

In how many ways can the group $\mathbb Z_5$ act on the set $\{1,2,3,4,5\}$? [closed]

In how many ways can the group $\mathbb Z_5$ act on the set $\{1,2,3,4,5\}$? A.5 B.24 C.25 D.120
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1answer
64 views

Non abelian group of order 40

Let $G$ be a non abelian group of order 40 that is the direct product of two of its Sylow subgroups. Show that the center of $G$ has order 10. I tried 40=5×8 Since $G$ not abelian $G$ NOT EQUAL ...
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1answer
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Find smallest $x$ such that $a^x \equiv b \bmod p$

Problem: How do we find smallest $x$ such that $a^x \equiv b \bmod p$, where $p$ is a prime and $1 \le b,a \le p$ and $a$, $b$, and $p$ are given and fixed. If there is no such $x$, how do we check ...
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48 views

abstract algebra question with p-subgroup

Let $G$ be a finite group, $H<G$ a subgroup. Suppose that $H$ is a $p$-group, $p$ a prime, and $p\mid[G:H]$. Show that $p\mid[N_G(H):H]$ and $H<N_G(H)$. In particular, if $G$ is a $p$-group and ...
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1answer
48 views

Vector Spaces and Groups

I've just completed a course in linear algebra. I'm a physics undergraduate and I don't plan on taking an abstract algebra course. That said, I've been reading a little bit about it. As I understand ...
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2answers
33 views

Understanding the definition of tensor product as a quotient of a free abelian group

I've been give the Definition: Let F be a free abelian group with a basis $X$ such that. $$F = \langle A\times B\mid \emptyset \rangle $$ Let $f$ be a subgroup of $F$ generated by the ...
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1answer
26 views

The Centralizer $C_H(x)$ where $x \in G$ and $H \leq G$.

Let $G$ be a group and $H$ be a subgroup of $G$. Let $x \in G$. Then $C_H(x)=H$ if and only if $x \in Z(H)$? It is obvious that if $x \in Z(H)$ then $C_H(x) = H$. But I could not prove or provide ...
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what is the “largest” abelian subgroup of SL(2,Z)?

I suppose "largest" is a bit of a nebulous term, so to make it precisely, I suppose I could ask for the largest size of a minimal generating set of an abelian subgroup's image in $PSL(2,\mathbb{Z})$. ...
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2answers
42 views

Order of Elements in Quotient Groups (Proof)

Let G be a group and N $\lhd$ of G such that [G : N] = 100. Suppose that a $\in$ G such that $a^{23}$ = e, where e is the identity element of G. Show that a $\in$ N. Since the order of the group G/N ...
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0answers
20 views

Complex numbers and circle group [duplicate]

Let $U_{1}$ be the circle group on $\mathbb{C}^\times$ For my mathimatics study, I have to prove that there is an isomorphism f: $U_{1}\times \mathbb{R}^+$ $\rightarrow$ $\mathbb{C}^\times$ How do ...
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1answer
27 views

Centralizer, Normalizer and Conjugate

I am looking at Group Theory notes on Centralizer and Normalizer for next semester and come up with this question: Let $H$ be a subgroup of $G$, and let $g$ be an element in $G$. Show that $$(a)\ ...
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isomorphism and circle group [closed]

For my mathimatics study, I have to prove this: Let $U_{1}$ be the circle group on $\mathbb{C}^\times$. Prove that $\mathbb{C}^\times$ is isomorphistic to $U_{1}\times \mathbb{R}^+$. How do I ...
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1answer
34 views

Prove homomorphism and surjectivity of a function

I have a question about this exercise for my math study: Let $d, n \in\mathbb{Z}_{>0}$ with $d\mid n$. a) Prove that there is a homomorphism $g: (\mathbb{Z}/n\mathbb{Z})^*\rightarrow ...
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Is it ever possible for hypercomplexes to generate every element modulo a prime?

To start, we can take a well-chosen complex number, modulo a prime $p$, and generate every complex element modulo $p$. For example, if we take $(1+2i)^k \pmod 3$, each power of $k$ up to $(3 \cdot 3 ...
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Motivation behind automorphism bases?

Given a model $\mathcal{M}$ with a domain $M$ and $B \subseteq M$, $B$ is an automorphism base for $\mathcal{M}$ iff $\forall f \in Aut(\mathcal{M}). (\forall b \in B. f(b)=b) \implies f = ...
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3answers
139 views

Prove a quotient group is abelian

Let $G$ be a group with a normal subgroup $M$ such that $G/M$ is abelian. Let $N\geq M$ and $N \unlhd G$. Show $G/N$ is abelian. My attempt: To show that $G/N$ is abelian, we need to show that for ...
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Show the intersection of a nonidentity normal subgroup and the center of P is not trivial

P is p-group and M is a nontrivial normal subgroup of P. Show the intersection of M and the center of P is nontrivial. By the class equation, I proved that Z(P)is not 1. Then, how do prove I the ...
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1answer
28 views

If $N \trianglelefteq G$ and $G$ is $\frac{3}{2}$-fold transitive, then $N$ is transitive or semiregular.

If $N \trianglelefteq G$ and $G$ is $\frac{3}{2}$-fold transitive, then $N$ is transitive or semiregular. I proved that if $N$ is intransitive, then $G$ will be imprimitive and Frobenius but I don't ...
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1answer
40 views

If $N \lhd G, (G : N) = 100, a \in G, a^{23} = e$, show that $a \in N$.

I'm having some trouble with a question that my instructor suggested to think about: Let $N \lhd G$ be a normal subgroup of $G$, the index $(G : N) = 100, a \in G, a^{23} = e$. Show that $a \in N$. ...
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1answer
35 views

Minimal number of relations for finite $p$-groups

From the (sharpened) Golod/Shafarevich inequality we know that for finite $p-$groups, where $r$ is the minimal number of relations and $d$ is the minimal number of generators, that $r > ...
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1answer
37 views

Five Cubes in Dodecahedron

I will demonstrate why the group of rotational symmetries of a Dodecahedron is $A_5$. For that, we have to find five nice objects, on which the group of symmetries acts. One object is "Cubes" ...
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8answers
363 views

Every infinite abelian group has at least one element of infinite order?

Is the statement true? Every infinite abelian group has at least one element of infinite order. I am searching for an infinite abelian group with all elements having finite order. Please help me ...
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2answers
52 views

Group action on set

Let $A=\{a,b,c,d\}$ be a set consisting of 4 distinct elements. In this question, a group action $s:\mathbb{Z}_4\to S_A$ is considered. Here $\mathbb{Z} _4=\{0,1,2,3\}$ while the group operation in ...
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1answer
38 views

homomorphism and resideu classes

I have a question about how I have to do this exercise for my math study: Let d, n $\in$ $\mathbb{Z}$>0 with d|n. a) Prove that there is a homomorphism f: $\mathbb{Z}$/n$\mathbb{Z}$ $\rightarrow$ ...
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3answers
62 views

Prove that $G$ is abelian iff $\varphi(g) = g^2$ is a homomorphism

I'm working on the following problem: Let $G$ be a group. Prove that $G$ is abelian if and only if $\varphi(g) = g^2$ is a homomorphism. My solution: First assume that $G$ is an abelian group ...
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1answer
58 views

Generalized Cauchy's theorem (group theory)?

I think there is a corollary for Cauchy's theorem such that a finite group $G$ contains a subgroup of order $n$ for each divisor $n$ of $|G|$ with a certain condition (I can't remember what it was). ...
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3answers
69 views

show that there exist an element $g$ of a group $G$ such that $g^q$ is in $H$

$H$ is normal in $G$ and $q$ is prime with $q\mid[G:H]$. Show that there exist $g$ in $G$ such that $g^{q}$ is in $H$. I am not sure how to use $q\mid[G:H]$. Should I try to show that $f:G→H$, with ...
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0answers
35 views

Cosets of a Normal Subgroup, Quotient Group

Consider the following subgroup of the symmetric group $S_4$: $V_4$ = {(1), (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)}. (a). Show that $V_4$ is a normal subgroup of $S_4$. (b). Find a permutation α ...
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1answer
56 views

Show that there are at least $2$ elements in $U(n)$ such that $x^2=1$.

I am working on some exercises in Joseph Gallian's Contemporary Abstract Algebra. I came upon the following: Show that there are at least $2$ elements in $U(n)$ such that $x^2=1$, for $n>2$ ...
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1answer
39 views

show a quotient group is cyclic

Let a and b be nonzero integers. Let $\sigma:\mathbb{Z}\times \mathbb{Z}\to\mathbb{Z}$ and $\sigma((x,y))=ax+by.$ Suppose $\gcd(a,b)=1$. Show that $(\mathbb{Z}\times \mathbb{Z})/\langle(a,b)\rangle$ ...
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2answers
54 views

Find all subgroups of group

Given multiplicative group of integers modulo 13: $Z_{13}^*$. Find all subgroups of this group. I need to proove, that this group is cycled. Also, as $|Z| = 12$, I know, that if $H$ is a subgroup of ...