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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Identity involving 2- and 3-cycles in group $S_3$

I would like to prove that a product of any 2-cycle $\sigma_2$ in symmetric group $S_3$ with any 3-cycle $\sigma_3$ in $S_3$ satisfies the identity $\sigma_2 \sigma_3 = \sigma_3^{-1} \sigma_2$. I don'...
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39 views

Group isomorphism between $D_3$ and $S_3$

If one wants to prove that $D_3$ is isomorphic to $S_3$, would it be sufficient to define a homomorphism $\psi: D_3\to S_3$ and argue that it is well-defined since $\psi(sr^i)=\psi(s)\psi(r)^i=\...
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71 views

Group of order $143$, counting argument.

I am trying the following exercise (I cannot use Cauchy's theorem, it's not in my course yet) : Let $G$ be a group of order $143=11\times13$, prove that that $G$ contains an element of order $13$...
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1answer
44 views

Better way to compute commutator subgroup of $A_n$ for $n\geq 5$

I want to show that $[A_n,A_n]=A_n$ for $n\geq 5$. Clearly $[A_n,A_]\leq A_n$. But to show the reverse inclusion I have an answer which involves too much "calculation"(As $A_n$ is generated by the ...
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45 views

Proving Every Principal Ideal Domain is Unique Factorization Domain

Im trying to learn the proof of Every Principal Ideal Domain is Unique Factorization Domain ...
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41 views

About Centralizers of two elements of a group.

Let $G$ be a group, and suppose $b$ belongs to $G$ with $|b|=7$. Prove that $C(b)=C(b^4)$. I'm not entirely familiar with centralizers, if someone could give me a push in the right direction, that ...
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2answers
61 views

Isomorphism between two groups mapping the same elements

I need to show that there is an isomorphic function $f:\Bbb Z_6 \to U(14)$ that $f(5)=5$ I can show that $U(14)$ is isomorphic to $\Bbb Z_6$ by showing that $U(14)$ is cyclic of order 6, but I do not ...
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1answer
36 views

Drawing a Cayley table given a binary operation - not sure how to proceed

Screenshot of the question ( I don't have enough rep points to post images) [Edit: then don't post images, post text. Here it is (with MathJax for you too).] Let $Y=\{\emptyset, \{\emptyset\}\}$ ...
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5answers
134 views

Let $G$ be a finite abelian group. Let $w$ be the product of all the elements in $G$. Prove that $w^2 = 1$.

The Statement of the Problem: Let $G$ be a finite abelian group. Let $w$ be the product of all the elements in $G$. Prove that $w^2 = 1$. Where I Am: Well, I know that the commutator subgroup of $G$...
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2answers
97 views

Groups containing two rotations.

Let $f$ and $g$ be rotations of the plane about distinct points, with arbitrary nonzero angles of rotation $\theta$ and $\phi$. Does the group generated by $f$ and $g$ contain a translation?
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85 views

To prove that element $\frac{3}{n}+i\frac{4}{5}$ has an infinite order in $\mathbb{C}$ for any $n\in\mathbb{Z}\backslash \{0\}$

The problem is to prove that element $z=\frac{3}{n}+i\frac{4}{5}$ has an infinite order in the group $(\mathbb{C},\, \cdot\, )$ for any non-zero integer $n$. Let's consider the case $|n|\neq 5$. The ...
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2answers
139 views

Proof that the following multiplicative groups modulo m are cyclic

In the Wikipedia page about the multiplicative groups modulo $m$, the following claim is made: The group $(\mathbb{Z}/m\mathbb{Z})^*$ is cyclic if and only if $m=1, 2, 4, p^k$ or $2p^k$, where $p$ is ...
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0answers
37 views

If $G$ acts such that each non-trivial element either has no fixed point or exactly two, then there exists fixed-point free involutory map on $\Omega$

If a finite group $G$ acts on a set $\Omega$ non-regulary (i.e. there is some element fixing some point) and each element having some fixed point has exactly $n$ fixed points, then we say the group is ...
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2answers
74 views

Group of automorphisms of $(\mathbb{R}, +)$

I understand that the group of automorphisms of $\mathbb{Q}$ (as a group) is isomorphic to $\mathbb{Q}^{\times}$. I am wondering what the group of automorphisms of $\mathbb{R}$ (as an additive group)...
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119 views

What equational properties of a group only need to be checked on a generating set?

Let $G$ be a group and $S\subset G$ a generating set. Let $P$ (short for $P(x_1,\dots,x_n) = 1$) be an equational property that may or may not be satisfied by all $n$-tuples of elements of $G$. My ...
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0answers
57 views

Theorem about non-regular group action, where an element fixes no points or exactly $p$ points

I have a question on the following proof. All groups are assumed to be finite. But first I will mention a lemmata: Lemma: Let $G$ act faithfully and non-regular as a group such that there exists some ...
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1answer
41 views

How do I prove that a group with one generator and a single relation is isomorphic to $\mathbb{Z_m}$?

That is, if $G= \langle a \, | \, a^m = e \rangle$, then $G$ is isomorphic to $\mathbb{Z_m}$.
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1answer
35 views

Order of a complex number

The complex number "$z$" has the same order as $(z^{p}*e)$ where $p$ is a prime and $e$ is a p-th root of unity. What are the possible values for the order of $z$?
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28 views

Linear space with basic vectors of outer space

In general, we set a n-dimensional linear space $\mathscr{L}$, then we would have n-dimensional basic vectors. Let's look at this in different way. We find the basic vectors from the outer space which ...
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44 views

Subgroups of $G=(\mathbb{Z}_{12},+)$

Draw the Hasse Diagram for the subgroup lattice of $G=(\mathbb{Z}_{12},+)$ Identify the subgroups $K=\langle [6]\rangle$ and $L=\langle [9]\rangle$ on your diagram. What are $K\cap L$ and $\langle K,...
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59 views

$\otimes$-Categorical Generalization of Lagrange

I am reading M. Brandenburg's paper and came across the following result which is a generalization of Lagrange's theory in group theory: Let $\mathcal C$ be a $\otimes$-category and $A\to B$ a ...
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1answer
52 views

The structure of groups (p-groups) in which the centralizer of any non-central element is cyclic

I would like to ask about the structure of non-abelian groups ($p$-groups) in which the centralizer of any non-central element is cyclic. Is there any classification about these groups?
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1answer
33 views

How to understand this statement?

Let $G$ be a group. For each element $g\in G$, $p_g$ is the map $G\rightarrow G$ defined by $p_g(x)= gx$ for all $x\in G$ Prove, that $p_n$ is a permutation of $G$ (regarded as a set). Does that ...
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1answer
47 views

Importance of $2$-groups in Finite Simple Groups

In a forward to a book on Groups of Prime Power Order, Z. Janko says Of special interest are $2$-groups. In fact, if $G$ is a non-abelian finite simple group and if the structure of its Sylow-$2$...
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1answer
43 views

Showing that $A_n$ is generated by the 3-cycles in $S_n$

I am trying to show that $A_n$ is generated by the 3-cycles in $S_n$. It seems that every 3-cycle of the form $(a_1,a_2,a_3)$ can just be written as $(a_1,a_3)(a_1,a_2)$ so every 3-cycle turns into an ...
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28 views

Let $\beta \in S_n$ be an $r$-cycle. How to show that $\beta \in A_n$ iff $r$ is odd?

Let $\beta \in S_n$ be an $r$-cycle. I am trying to show that $\beta \in A_n$ iff $r$ is odd. By assumption we have $\beta = (a_1,a_2,a_3,...,a_r)$ for some $r \in \mathbb{N}$. Then assuming $\beta \...
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43 views

Proving whether the following are groups or not.

In each case, I am asked to decide whether the indicated pair is a group or not. If so, prove it; if not, show which group axiom fails. (a) $(\dfrac{1}{2}\mathbb{Z}, +)$ where $\dfrac{1}{2} \mathbb{Z}...
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49 views

Group generated by two plane rotations containing a translation

Given a group generated by two rotations of the plane about distinct points, how to show that it contains a translation? Attempt: Let the two rotations be $f$, $g$, then consider the motion $f \circ ...
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44 views

Finding the cycle decomposition of a given permutation.

I am given the following permutation and need to find its cycle decomposition: $\left(\begin{array}{ccccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 7 & 5 & 8 &...
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1answer
28 views

Are there any examples of groups with a 4-regular cayley graph with an euler characteristic of 2?

The motivation here is that I'm looking for a group that topologically (based on its Cayley Graph) behaves like a sphere, but algebraically is similar to the direct product of two cyclic groups (which ...
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1answer
25 views

Prove that $ A^n := \{ a^n : a \in A \}$ and $A_n := \{ g: g \in A : g^n = 1 \} $ are characteristic subgroups of $A$.

The Statement of the Problem: Let $A$ be an abelian group and $n$ a positive integer. Prove that $$ A^n := \{ a^n : a \in A \} $$ $$ \text{and} $$ $$ A_n := \{ g: g \in A : g^n = 1 \} $$ are ...
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42 views

Is the center of an Abelian group every element in the group?

If $G$ is an Abelian group then $Z(G)$ is the set that contains every element in $G$. I'm not sure if this is true and if it is how to go about showing it. Appreciate any help.
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48 views

Can a subgroup be not equal to the group?

Let A be the group of positive rational numbers under multiplication. Prove that A is isomorphic to a subgroup B such that A does not equal B. So I know how prove isomorphisms.(define mapping, prove ...
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Direct Product and Quotients/Factor groups

Let $A, B \le G$ subgroups and $N \unlhd G$ a normal subgroup of $G$, then of course \begin{align*} AN/N \cdot BN/N & = \{ aN \cdot bN : aN \in AN/N, bN \in BN / N \} \\ & = \...
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2answers
62 views

Finding the commutator subgroup of $S_3$ and the commutator subgroup of that commutator subgroup.

I'm finding this idea very frustrating. I need to find the commutator subgroup of $S_3$ (call it $S_3'$) and the commutator subgroup of that commutator subgroup (call it $S_3 ''$). So, I have that, ...
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2answers
62 views

$\mathbb{R}/H$ has a nontrivial element of finite order

I am currently struggling with this problem: Let $H$ be a nontrivial subgroup of $\mathbb{R}$. Prove that $\mathbb{R}/H$ has a nontrivial element of finite order. I believe this refers to the ...
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1answer
31 views

Why is this proof incorrect? (Group Theory)

If $G$ is a finite non-trivial group and $H \leq G$ has index $p$ prime, then $H \lhd G$. This statement is actually false. However, I cannot find the error in what I wrote. Let $G$ act on the ...
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1answer
39 views

Proving that the multiplicative group modulo $2^r$ are cyclic iff $r<3$

No idea how to start, can someone give me a hint? Show that $(\mathbb{Z}/2^r\mathbb{Z})^*$ is a cyclic group if and only if $r<3$.
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16 views

Common eigenvalue for a unipotent group of $GL_n (K)$ in positive characteristic

If we set $G$ a unipotent sub-group of $GL_n (K)$ with $car(K)>0$ ($\forall g\in G\quad g=1+n$ where $n$ is nilpotent), we wish to prove that $G$ is conjugate to a sub-group of $T$, the group of ...
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33 views

Abelian p-group

Let $G$ be a group such that for all $a$, $b$ $\in$ G, there exists $f$$\in$ Aut(G) satisfy $b$=$f(a)$. i) Prove that if there is nontrivial élément in $G$ of finite order, then there exist prime ...
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1answer
56 views

Generating set of the real numbers under addition

How would one generate $(R,+)$? If you took an arbitrarily small interval, would this work? How can you prove that this is so?
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1answer
73 views

Let $G'$ be the commutator subgroup of a group $G$. Prove that $G'$ is the intersection of all the $K\mathrel{\unlhd}G$ for which $G/K$ is abelian.

The Statement of the Problem: Let $G'$ be the commutator subgroup of a group $G$. Prove that $G'$ is the intersection of all the $K\mathrel{\unlhd}G$ for which $G/K$ is abelian. Where I Am: I feel ...
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1answer
46 views

$H, N$ subgroups of $S_{5}$

This is a homework problem but I have stuck at some other chapter earlier so I am now completely lost what I am supposed to do. Clues and hints and suggestions of theorems that I should look up would ...
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0answers
27 views

If $ G $ is a finite soluble group and satisfies the permutizer condition, then for any odd prime $ p $, $ G $ is $ p $-supersoluble.

If $ G $ is a finite soluble group and satisfies the permutizer condition, then for any odd prime $ p $, $ G $ is $ p $-supersoluble. It is for me problem that why $ G $ can't $ 2 $-supersoluble group....
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26 views

Prove that every group (V, $*$, I) satisfies I^-1 = I

Prove that every group (V, $*$, I) satisfies I$^-$$^1$ = I. According to my book, the solution is: Take the definition of x$^-$$^1$: x $*$ x$^-$$^1$ = I $\land$ x$^-$$^1$ $*$ x = I Then, by using ...
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write down a concrete 2-coboundary

Let $G=SL_3(\mathbb{Z})$, the group of 3 by 3 matrices with determinant 1. Then by a deep theorem of Borel and Serre proved in this paper "Corners and arithmetic groups", we know that the 2nd ...
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3answers
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Is my proof correct? (group theory, Lagrange)

$G$ is a finite group. Prove, that $\exists k\in G:k\neq 1_G, k^2=1_G \iff |G|$ is even. My proof 1) $\Rightarrow$: $\exists k\in G: k^2 = 1_G\Rightarrow \exists G'\leq G$ s.t. $ G'=\{1_G,k\}$, ...
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1answer
41 views

Proving that $U(n)\subset SO(2n)$

I have to prove that $\text U(n)$ is a subgroup of the group $\text{SO}(2n)$. What I have tried is just splitting a vector $x\in\mathbb C^n$ in a vector in $\mathbb R^{2n}$ by just separating the ...
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1answer
49 views

Group Theory, conjugation of permutation group

I've been given the following question in the context of group actions through conjugation but I'm having difficulty understanding what is being asked Let $\tau$ be any permutation in $S_m$. Let $\...
8
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1answer
114 views

Does $A\oplus \mathbb{Z}\cong B\oplus \mathbb{Z}$ imply $A\cong B$?

If $A$ and $B$ are abelian groups, do we have that $A\oplus \mathbb{Z}\cong B\oplus \mathbb{Z}$ implies $A\cong B$? Motivation: I was just thinking about different ways of deducing equality from ...