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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1answer
57 views

Why $PGL(2, 9)$ is not isomorphic to $S_6$?

How can I show that $PGL(2,9)$ is not isomorphic to $S_6$? My primary idea is to compare the size of conjugacy classes of two well-chosen elements in these groups. Is there another simpler ...
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1answer
47 views

how to deduce that finite group $G$ has at most $|G|$ characters

Let $G$ be a finite abelian group. A character of $G$ is a group homomorphism $\chi: G\longrightarrow \mathbb{C}^{\times}$. I have proved by induction that for distinct $\chi_1,\cdots,\chi_r$, they ...
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3answers
76 views

Prove every isometry of the plane is expressible as product of reflections, translations, rotations

Prove every isometry of the plane is expressible as product of reflections, translations, rotations I know that the distance preserving isometries are a group but I have no idea how to use this ...
4
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1answer
118 views

For any $n$, there are at most two simple groups of order $n$? [duplicate]

How do you prove that for any $n$ there are at most two simple groups of order $n$?
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1answer
112 views

Constructing a rotation matrix from complex eigenvalues

I am trying to construct a rotation matrix $\mathbf{R}\in\mathbb{R}^{3\times3}$ rotating around an axis $\hat{n}$ in a basis $\{\hat{n},\hat{u}_{1},\hat{u}_{2}\}$. Formally: Given a basis ...
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1answer
141 views

Showing that every subgroup of a metacyclic group is metacyclic

I have tried to prove that every subgroup of a metacyclic group is metacyclic (where a group $G$ is metacyclic if it has a normal cyclic subgroup $C$ such that $G/C$ is cyclic). I believe I have a ...
2
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1answer
112 views

Isomorphism and direct product of groups

I need some clarification. Consider these two groups: $\mathbb{Z}_{m}\times\mathbb{Z}_{k}$ and $\mathbb{Z}_{n}$, where $mk=n$. I know that if $m= 1$ or $k= 1$, then ...
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1answer
127 views

If $X$ is $G$-paradoxical then $G$ is $G$-paradoxical. Is my proof correct?

I am currently reading Stan Wagon's Banach-Tarski Paradox book, and this was left as an exercise to prove (converse of Proposition 1.10). Let $X$ be a set, and let $G$ act on $X$ with no ...
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2answers
115 views

If a group has a faithful reducible two-dimensional representation then its commutant is abelian.

A group $G$ has a faithful reducible two-dimensional representation. Prove that commutant of the group $G'$ is Abelian. I think to so. Commutant $G'\triangleleft G$. Let $\rho$ is the faithful ...
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0answers
56 views

Lie algebra of the unipotent radical of a standard parabolic subgroup in $GL_n$

Let $k$ be a field, and consider the algebraic group $G=GL_n(k)$. For any partition $n_1+n_2+\ldots+n_m=n$, we have a parabolic subgroup of the form ...
10
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1answer
254 views

Intuition about the class equation (and flowers).

I apologize in advance for the size of the images I've devoted a lot of time and effort to draw them on the computer and i didn’t manage to re-size. I'd be really thankful if anyone would edit this ...
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2answers
87 views

Is a set that is an abelian group under addition and a group under multipliation a field?

I suspect the answer to my question is yes, but I'm just checking my understanding. If we have a set which is an Abelian group under addition and a group under multiplication is it then defined as a ...
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1answer
135 views

Prove a group is an abelian group

Let $G\subseteq \mathbb N.$ How do I prove that $G$ is an abelian group with respect to the binary operation " * " defined by $\;a*b = a+b+11$ ?
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1answer
36 views

Question about question about finite order elements at $\mathbb{C}^*/U$

At this question - What are element with finite order at $\mathbb{C}^*/U$? I understand that finite order at $\mathbb{C}^*/U$ are only the $e$ elements. Now, I have two questions: It is because ...
7
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1answer
85 views

Proof of $|C_{G/N}(gN)| \leq |C_G(g)|$ without character theory

In the book Character Theory of Finite Groups by Isaacs the following is proven (Corollary 2.24) using character theory: Proposition: Let $G$ be a finite group and $N \trianglelefteq G$. Then ...
3
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3answers
111 views

Are $\Bbb Z_{8} \times \Bbb Z_{10} \times \Bbb Z_{24}$ and $\Bbb Z_{4} \times \Bbb Z_{12} \times \Bbb Z_{40}$ isomorphic? [closed]

Are the groups $\Bbb Z_{8} \times \Bbb Z_{10} \times \Bbb Z_{24}$ and $\Bbb Z_{4} \times \Bbb Z_{12} \times \Bbb Z_{40}$ isomorphic? Why or why not? (Here $\times$ means the direct product or direct ...
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0answers
43 views

“Fixed Domains” of a Linear Transformation

Given a linear transformation $T$, I need to find the set of all domains $D$ such that $T:D\mapsto D$. Equivalently, I need to find the set of all domains $D$ that are symmetric under $T$. Aside from ...
4
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2answers
127 views

Generalizing central automorphism group condition to endomorphisms

Given a group $G$, we can define its central automorphism group by $$\operatorname{Aut}_c(G)= C_{\operatorname{Aut}(G)}(\operatorname{Inn}(G)) = \{ \phi\in\operatorname{Aut}(G) : \phi(g)g^{-1} \in ...
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3answers
127 views

Subgroup of a symmetric group $S_7$

Is there any quick way to determine if $S_7$ contains a subgroup of order $6$ or $S_{11}$ a subgroup of order $30$ ? A problem carrying only 2 marks involves this. So I assume either there is some ...
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3answers
112 views

Extend a function to group homomorphism

This is maybe a trivial question. Set up Assume $S = \lbrace g_{1}, \dots, g_{n} \rbrace$ generate a group $G$ and $H$ is a finite group with elements $\lbrace h_{1}, \dots, h_{n} \rbrace$. In ...
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1answer
156 views

Presentation of Heisenberg Group $\mathbb{H}$ over the field $\mathbb{F}$

Let $\mathbb{F}$ denote the finite field. Denote $\mathbb{H}_{\mathbb{F}}=\left\{ \left( {\begin{array}{ccc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \\ \end{array} } ...
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votes
1answer
68 views

Getting free groups of finite/countable rank from certain generating sets

Since I don't know exactly how to explain this I will first describe the idea with the free group on one generator (which I will treat as the integers). Lets say you are given an infinite sequence of ...
6
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2answers
161 views

Are there groups $G$, whose order is uncountable, such that the order of $G/[G,G]$ is countable?

Are there groups $G$, whose order is uncountable, such that the order of $G/[G,G]$ is countable? I am mostly concerned with looking at the groups in terms of generators and relations, so this can be ...
0
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1answer
124 views

Prove the theorem about cardinality of subgroup [duplicate]

$G$ is a group. If $H, K \le G, |H|, |K| < \infty$, then $$|HK| = \frac{|H||K|}{|H \cap K|}$$ How to prove it? UPD: I tried to prove that this function is bijective: So if i prove it, I ...
3
votes
2answers
105 views

What is the meaning of $\bigsqcup$?

This is the expression: $$G=\bigsqcup_{d\mid n}X_d$$ I saw it here - Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic. Thank you!
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3answers
58 views

classification of $2$-dimensional field extensions

Let $F$ be a field and $K:F$ be a field extension such that $[K:F]=2$. Then (i). If $Char(F)\neq 2$, then there exists $\alpha\in K^*$, $\alpha\notin F^*$, such that $K=F(\alpha)$ and $\alpha^2\in ...
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0answers
60 views

Proving a theorem at groups theory [duplicate]

$G$ is a finite group. Let assume that for every $m\in \mathbb{N}$ there are maximum $m$ elements s.t. $x^m=e\;$. $(x\in G)$ I need to prove that $G$ is cyclic group. I need to use the fact that: ...
2
votes
4answers
265 views

example of Frobenius endomorphisms that is not automorphism

Let $F$ be a field of characteristic $p$ and $F$ has infinitely many elements. Prove that the map $\sigma: \alpha\mapsto \alpha^p$ is a field endomorphism but not necessarily automorphism. How to ...
3
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3answers
164 views

On the 11 Sylow subgroup of a group of order 792

It seems that the 11-Sylow subgroup of a group of order $792=11×2^3×3^2$ is normal. Could you help me why it is true
3
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2answers
60 views

What are element with finite order at $\mathbb{C}^*/U$?

I need to find the elements with finite order at the group - $\mathbb{C}^*/U$. $U$ - is the Circle Uint. $\mathbb{C}^*$ - is $(\mathbb{C}/0,\cdot)$. I need to write also the proof, and I'll be glad ...
2
votes
0answers
30 views

Linear group action over an hermitian space.

Let $\mathrm{h}$ be an non degenerate hermitian form on $\mathbb{C}^2$, and suppose $\mathrm{h}$ is of signature $(1,1)$. Let $A$ and $ B \in \mathrm{SL}(2,\mathbb{C})$ such that $G$ the group ...
2
votes
3answers
88 views

What are the finite order elements of $\mathbb{Q}/\mathbb{Z}$?

I need to find what are the at the group $\mathbb{Q}/\mathbb{Z}$. I think that any element at this group has a finite order, but I don't know how to prove it... I'd like to get help with the proof ...
13
votes
2answers
271 views

Is $SO_n({\mathbb R})$ a divisible group?

The title says it all ... Formally, if $SO_n(\mathbb R)=\lbrace A\in M_n({\mathbb R}) |AA^{T}=I_n, {\sf det}(A)=1 \rbrace$ and $W\in SO_n(\mathbb R)$, is it true that for every integer $p$, there is ...
1
vote
1answer
398 views

Prove that if a finite solvable group is simple, it is a cyclic group of prime order. [closed]

Prove that if a finite solvable group is simple, it is a cyclic group of prime order. Help me some hints.
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votes
3answers
206 views

What is this group? (Recognising a group from a presentation).

I am trying to find out what the following group is: $$G = \langle a, b \mid ab^2 = b^2a,\ a^4 = b^3\rangle.$$ Due to the isomorphism problem for groups, there is no algorithmic way to approach ...
4
votes
1answer
218 views

Determine Galois groups of polynomials

Determine the Galois groups of the following polynomials over Q. $f(x)=x^3−3x+1$. $g(x)=x^4+3x+3$. $h(x)=x^5+8x+12$. I have no way to find their Galois groups. I only obtained that since they are ...
4
votes
0answers
131 views

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 $pq^3$. ...
0
votes
1answer
47 views

Find the number of inequivalent two-dimensional complex representations of the group $Z_4$

Find the number of inequivalent two-dimensional complex representations of the group $Z_4$ Any hints will be greatly appreciated. Thank you all
1
vote
1answer
80 views

locating a square root element with square in a free subgroup

Let $G$ be a group (not necessarily countable discrete) which contains a free subgroup $F=F_2=\langle a, b\rangle$, denote $H=\langle a, b^2\rangle$, and assume that the centralizer of $F$ inside of ...
4
votes
2answers
100 views

What can we learn purely from the existence of a (non-constant) functor to the category of abelian groups?

I admit that the following is a very broad question. So if you feel that it is too vague please say so. It might also just be that I haven't read enough about category theory and my question is silly. ...
0
votes
1answer
354 views

Need help with proving that group is not finitely-generated [duplicate]

I need to prove that $(\mathbb{Q}^*, \times)$ (i.e rationals, zero excluded, under multiplication) is not finitely generated. So, suppose that G is finitely-generated. That means there exist a ...
5
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0answers
160 views

$(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ is a cyclic group

I would like to prove that the group $(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ of the invertible elements of $\mathbb{Z}/p^r\mathbb{Z}$ with $p>2$ prime and $r>0$ is cyclic. My text suggests to ...
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2answers
112 views

References in Sylow's theorems, solvable groups, nilpotent group and Frattini subgroups

I'm very interested in group theory, i.e Sylow's theorems, solvable groups, nilpotent group and Frattini subgroups. Can anyone tell me some articles, textbooks, notes ... about them? Thanks a lot.
2
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0answers
70 views

Suppose that both $N$ and $G/N$ are nilpotent. Let $K$ be a nilpotent subgroup of maximal order, such that $G=NK$. Prove that $N_{G}(K)=K$.

Let $G$ be a finite group, $N$ be a normal subgroup of $G$. Suppose that both $N$ and $G/N$ are nilpotent. Let $K$ be a nilpotent subgroup of maximal order, such that $G=NK$. Prove that $N_{G}(K)=K$. ...
2
votes
0answers
45 views

Prove that $G/N$ is a nilpotent group of class at most $c$ iff $C_{c+1} \subset N$.

Let $C_{i}=C_{i}(G)$ be the higher commutator subgroups of a group $G$. Let $N$ be a normal subgroup of $G$. Prove that $G/N$ is a nilpotent group of class at most $c$ iff $C_{c+1} \subset N$. I try ...
2
votes
1answer
114 views

Prove that if both $N$ and $G/N'$ are nilpotent, then $G$ is nilpotent.

Let $G$ be a group, and let $N$ be a normal subgroup of $G$. Let $N'$ be the derived group of $N$. Prove that if both $N$ and $G/N'$ are nilpotent, then $G$ is nilpotent. Furthermore, assume that $G$ ...
4
votes
2answers
78 views

Uniqueness of Perfect group of a given order?

Is there only one perfect group (up to isomorphism) of a given order? My intuitive thought is that a perfect group has such stringent requirements on the group product that it must be unique. I do ...
0
votes
3answers
112 views

Is this identity for the Dihedral group correct?

Let $D_n$ represents the Dihedral group with $2n$ elements, and my question(based on some physics backgrounds) is: Does $Z_2$ a normal subgroup of $Q_8$? If it is, then is the indentity $D_2\cong ...
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6answers
622 views

Generic elementary group theory problems.

This question is about generic group theory problems. here are examples for what I’m referring to: Prove that any group of order $p^2$, where $p$ is a prime, is abelian. Let $G$ be a ...
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0answers
80 views

Irreducible and regular representation

I was working on representations of $S_3$ and thought about the following problems: The symmetric group $S_4$ acts on $\mathbb{C}^{4}$ by permuting coordinates. Decompose this representation into ...