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

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66 views

Must the centralizer of a non-identity element of a group be abelian?

Q1: Must the centralizer of a non-identity element of a group be abelian? I challenge everyone by asking further about this question one step further. The definition of centralizer is: Let a be ...
6
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3answers
55 views

Automorphisms of $\mathbb{Z}/p\oplus\cdots\oplus \mathbb{Z}/p$

Consider the abelian group $$G = \underbrace{\mathbb{Z}/p\oplus\cdots\oplus \mathbb{Z}/p}_{n},$$ where $p$ is prime and $1\le n \le p$. I want to show that $G$ has no automorphism of order $p^2$. I ...
2
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1answer
24 views

When is a power of $m$ cycle is also an $m$ cycle?

I have a question taken from Abstract Algebra by Dummit and Foote ($pg.33$ $q.11$): Let $\sigma\in S_{n}$ be an $m$ cycle. Show that $\sigma^{k}$ is also an $m$ cycle iff $(k,m)=1$ My efforts: By ...
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0answers
18 views

quasiprimitive non-solvable groups

I'm looking for the reference about quasiprimitive unsoluble groups. Actually we can find a lot of useful things about quasiprimitive solvable groups in "Representations of solvable groups by Manz and ...
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3answers
556 views

What does it mean for a group to be Abelian?

I'm revising questions on groups for exams, and I still can't quite understand what an Abelian group is. Please help me understand, if anyone could give me a more simple explanation.
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2answers
49 views

Normal subgroup created by a bunch of elements

if I have a finite group $G$ and a bunch of elements that are the elements of a set $A$. How can I systematically calculate the smallest normal subgroup of $G$ that contains $A$? I am rather ...
4
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2answers
85 views

In group theory, is it true that $f(X \vee Y) = f(X) \vee f(Y)$?

($\vee$ denotes join). Let $G$ and $H$ denote Abelian groups, $X$ and $Y$ denote subalgebras of $G$, and let $f : G \rightarrow H$ denote a homomorphism. Then: $$f(X \vee Y) = f(X+Y) = f(X)+f(Y) = ...
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1answer
20 views

Schur Multiplier of general linear group

Ideally I would like to know the Schur multiplier of $Gl(n, F_3)$, but perhaps this is not reasonable to ask. But for a small fixed $n$, this should be known, but i could not find any result when ...
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1answer
35 views

Is my proof correct? (about commutators)

I want to prove the following fact: Let $G$ be a finite group and let $x, y\in G$ such that $[x, y] \in Z(G)$. Then $[x, y^s] = [x, y]^s$ for every $s\in \mathbb{Z}$. If we assume that $[x^r, y^s] ...
3
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0answers
41 views

Another Presentation of Certain Cyclic Groups

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^n\rangle $$ is cyclic of order $3(n+1)$, for $n=0 \mod 3$ or $n= 1 \mod 3$, $n\ge 0$. This ...
4
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2answers
194 views

Cyclic Group Presentation

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^7\rangle $$ is cyclic of order 24. This presentation was obtained using the Todd-Coxeter ...
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0answers
19 views

Homeomorphism between SU(4) and SO(6)

http://www.mat.univie.ac.at/~westra/so3su2.pdf said that $\mathrm{SU}(2)$ acts homeomorphism to $\mathrm{SO}(3)$, via $$ \begin{pmatrix} z & w \\ -\bar w & \bar z \end{pmatrix} \mapsto ...
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0answers
20 views

Homomorphic image of intersection equals intersection of homomorphic images?

This is a tangent of this question. I wanted to remark in my answer that it is not generally true that given a group homomorphism $f:G\rightarrow H$ and two subgroups $X,Y\leq G$ that $$f(X\cap ...
3
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4answers
114 views

Examples of Infinite Simple Groups

I would like a list of infinite simple groups. I am only aware of $A_\infty$. Any example is welcome, but I'm particularly interested in examples of infinite fields and values of $n$ such that ...
2
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2answers
40 views

Determine the center of this finitely presented group.

Consider the group $G(n)= \langle a, b \ \vert\ aba^{-1}=b^{n+1}, bab^{-1}=a^{n+1} \rangle$, $n\ge 1.$ Show that the center $Z$ is cyclic of order $n$ and that $G/Z$ is abelian of order $n^2$. This ...
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1answer
37 views

Join of finitely many nilpotent subgroups (+ additional properties) is nilpotent?

I have the following: A group $X$ and a family $(X_n)_{n \in \mathbb{Z}}$ of subgroups of $X$, such that the following holds: $X = \langle X_n \;\vert\; n \in \mathbb{Z} \rangle$ $X_n \times X_{n+1} ...
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1answer
49 views

Show that the symmetric group $S_p = <\sigma , \tau >$, where $\sigma$ is any transposition and $\tau$ is any p- cycle and p is a prime number.

Let $\sigma = (a_1\ a_2) \ \ and \ \ \tau = (a_1\ b_2\ \ldots\ b_p)$. (We have $a_2 = b_i$ for some i.).We know that $S_p$ is generated by $\{ (a_1\ a_2) \ \ and \ \ (a_1\ a_2\ \ldots\ a_p) \}$. So ...
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0answers
35 views

A finite group with some prescribed subgroup structure

Can you display a group $G$ satisfying the following condition? You can choose an element $g\in G$ such that the set $\{H\trianglelefteq G \colon H\ \text{does not contain any power of } g\}$ has ...
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1answer
42 views

Can every group be faithfully represented as a group of permutations?

Definition (Group action) An action of a group $G$ on a mathematical object $X$ is a group homomorphism $G \rightarrow \mathrm{Sym}(X)$. i.e. Given an action $f$ of a group $G$ on a mathematical ...
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1answer
31 views

My question is about the definition of a map called the “reduction map”.

Let $G$ be a group and $N$ normal in $G$. I have read about a map $\alpha : G\rightarrow \frac{G}{N}$ called the reduction map mod $N$. I would love if someone could please explain this to me. Is it ...
5
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1answer
46 views

Is S a group under matrix addition

Another matrix question! Let $$S=\{A \in M_2(\mathbb{R}):f(A)=0\}\text{ and }f\left(\begin{bmatrix}a&b\\c&d \end{bmatrix}\right)=b$$ Is S a group under matrix addition. Either prove that ...
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1answer
88 views

Group of order 1183 is abelian if and only if contains an element of order 91

Let $G$ be a group such that $|G|=1183=7\cdot 13^2$. Show that $G$ is abelian if and only if $G$ has an element of order $91=7\cdot 13$. What i did: $7||G|\Rightarrow \exists x\in G : |x|=7$ and ...
5
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1answer
59 views

Almost pointwise inner automorphism of free products of groups.

Let $A,B$ be groups, let $G = A\ast B$ be their free product and let $\phi \in \text{Aut}(G)$ be a automorphism of $G$. We say that $\phi$ is pointwise inner if $\phi(g) \sim_G g$ (there is $w \in G$ ...
2
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1answer
64 views
+100

The definition of the right regular representation

I'm having difficulties understanding the definition of the right regular representation as it appears in Dummit & Foote's Abstract Algebra text. On page 132 it says Let $\pi:G \to S_G$ be the ...
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0answers
27 views

The existence of a normal subgroup with finite index [duplicate]

Let $G$ be a group and $H$ be a subgroup of $G$ such that $[G:H]$ is finite , then how to prove that there is a subgroup $N \subseteq H$ , such that $N$ is normal in $G$ and $[G:N]$ is finite ?
4
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82 views

Factorization of a finite group by two subsets

I want to write a GAP program for checking the following question. Let $G$ be a given finite group with order $n$. Is it true that for every factorization $n=ab$ there exist subsets $A$ and $B$ ...
6
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1answer
47 views

Group elements $x$ and $y$ satisfying $x^2 = y^2x^2y$ and $yx^{-1}y^2 = x^7$ commute.

The Question Suppose that $x$ and $y$ are elements of a group such that $$x^2 = y^2x^2y$$ and $$yx^{-1}y^2 = x^7.$$ Show that $x$ and $y$ commute. Motivation This came up in another question, where ...
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0answers
30 views

Family of equivalent unitary representations is not a set.

I have recently come across a statement in the book: Kazhdan's property (T) by B. Bekka, P. de la Harpe, A. Valette at the beginning Appendix F.2. Fell topology on sets of unitary representations. ...
1
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1answer
90 views

Itzykson-Zuber integral over orthogonal groups

I would like to know is there a closed form expression for the following Itzykson-Zuber integral for the orthogonal case. $I=\int_{\mathcal{O}(p)} ...
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0answers
68 views

Why are Lie Groups so “rigid”?

This is probably a naive question, but here goes. To motivate my question, I'll consider a unit circle in $\mathbb C$ or $\mathbb R^2$. This is a compact Lie group equipped with the usual exponential ...
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2answers
44 views

How many recursively definable groups are there on $\mathbb{N}$?

How many non-isomorphic, (non-free), non-trivial, recursively definable groups are there on $\mathbb{N}$? I know we can at least get 1. Let $F:\mathbb{N} \to \mathbb{Z}$ be the "natural bijection". By ...
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1answer
49 views

what are the other 2 nontrivial elements of the automorphism group of $\Bbb Z/5\Bbb Z$?

It is known that the automorphism group of the units of $\Bbb Z/5\Bbb Z$ is isomorphic to the cyclic group of order $4$, so the automorphism group must also have $4$ elements. The two nontrivial ones ...
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6answers
1k views

If $x^m=e$ has at most $m$ solutions for any $m\in \mathbb{N}$, then $G$ is cyclic

Fraleigh(7th ed) Sec10, Ex47. Let $G$ be a finite group. Show that if for each positive integer $m$ the number of solutions $x$ of the equation $x^m=e$ in $G$ is at most $m$, then $G$ is cyclic. ...
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2answers
48 views

Rank of a free group

I am trying to know whether the following result is true. Let $F$ be a free group with a basis $X$, and let $X'=\{xF':x\in X\}$, where $F'$ is the commutator subgroup of $F$. Then, $|X|=|X'|$. ...
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1answer
26 views

Prove $Q_8$, the group generated by two complex matrices $A$ & $B$ (see below) is a nonabelian group of order 8.

Problem: Let $Q_8$ be the group (under ordinary matrix multiplication) generated by the complex matrices $A=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ and $B=\begin{pmatrix} 0 & i \\ i ...
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0answers
26 views

Separability of conjugacy classes in conjugacy separable semidirect products.

We say that group $G$ is conjugacy separable if for every $g \in G$ the set $g^G = \{cgc^{-1} \mid c \in G\}$ is closed in the profinite topology on $G$, i.e. for every $f \in G \setminus g^G$ there ...
2
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1answer
34 views

Please check my proof on: $\sim$ is an equivalence relation $\Leftrightarrow S<G$

Problem: Let $\emptyset\ne S\subset G$, where $G$ is a group, and define a relation on $G$ by $a\sim b\Leftrightarrow ab^{-1}\in S$. Show that $\sim$ is an equivalence relation if and only if $S$ is a ...
2
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1answer
37 views

Semidirect products as (amalgamated) free product

It is well known that $\mathbb{Z}\rtimes \mathbb{Z}_2$ is free product $\mathbb{Z}_2 \star \mathbb{Z}_2$. Are there more examples of these kind?
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1answer
25 views

Countably generated group has at most countably many finite index subgroups

I know that if $G$ is a finitely generated group, then $G$ has at most countably many finite index subgroups. Is this result still true if $G$ is countably generated?
2
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1answer
27 views

Why does $x^{m \cdot 2^i} \equiv -1$ with odd $m$ imply that $x$ has order $m \cdot 2^{i+1}$?

It is clear that $$x^{m \cdot 2^{i+1}} \equiv 1$$ for odd $m$ but is there a theorem or an obvious reason why $x$ cannot have order smaller than $m \cdot 2^{i+1}$? Context: I am trying to understand ...
0
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0answers
91 views

Prove $H_1 \cap H_2 \le H_1 $ when $H_1, H_2 \le G$ and $H_1$, $H_2$ are finite.

Prove that $H_1 \cap H_2 \le H_1 $ when $H_1, H_2 \le G$ and $H_1$, $H_2$ are finite. What I've done Use the definition of subgroup: $G$ is a group and $H \subseteq G$. $H \le G \iff HH=H $ and ...
0
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3answers
62 views

If $H$ and $K$ are subgroups of G then $H \times K$ is a subgroup of $G \times G$

I know that if $H$ and $K$ are subgroups of $G$ then $HK= \{ hk \mid h \in H , k \in K\}$ is not necessarily a subgroup of $G$, this requires that $HK = KH$. But it follows that if $H$ and $K$ are ...
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2answers
266 views

Is there a group-theoretic proof of the Riemann rearrangement theorem?

The analytic proofs of the Riemann rearrangement theorem are easy to understand but they don't explain why commutativity breaks down when you go from finite sums to infinite sums of real numbers. I ...
2
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1answer
76 views

How to prove “a group $G$ of order $72$ can't be a simple group”?

By using Sylow theorem, I can prove that $G$ has either $1$ Sylow $3$-subgroup or $4$ Sylow $3$-subgroup, but I don't know how to continue the proof.
4
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2answers
77 views

Realizing $\mathbb{Z}/n \mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}$ as an isometry group

Every finite group arises as the isometry group of a subspace of an euclidean space (Albertsona, Boutin, Realizing Finite Groups in Euclidean Space). What are natural examples of spaces realizing the ...
2
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1answer
41 views

Lower central series and the center of $S/S^k$

Let $S$ be a $p$-group and assume that $S$ has maximal class - so if we assume $\vert S \vert = p^n$ then the lower (and upper) central series has length $n-1$. I don't know much about lower central ...
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0answers
50 views

Give a examples of $p$-Sylow subgroups of $G=A_5\times S_3$ for $p=2,3,5$ and tell what group is isomorphic to each subgroup? [on hold]

Give a examples of $p$-Sylow subgroups of $G=A_5\times S_3$ for $p=2,3,5$ and tell what group is isomorphic to each subgroup? Does there exist a subgroup of $G$ of order $180$?
1
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1answer
47 views

Irreducible action of a group on a set

Let $p$ be a prime. I am solving a problem and I am told that I should use that the action of $\text{SL}_2(p)$ on $\mathbb{F}_p^2$ is irreducible. But I don't know what this means?
3
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2answers
387 views

Is isomorphism not always unique?

Given two isomorphic groups G and H, is it possible that two or more functions define their isomorphism? Also, is it possible that another group say, L is isomorphic to G but not to H?
4
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1answer
74 views
+100

Is there a finite non-solvable group which is $p$-solvable for every odd $p\in\pi(G)$?

Let $G$ be a finite non-solvable group and let $\pi(G)$ be the set of prime divisors of order of $G$. Can we say that there is $r \in \pi(G)-\{2\}$ such that $G$ is not a $r$-solvable group?