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

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5
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1answer
20 views

When are two direct products isomorphic

I was thinking about the following problem: Suppose that $G_1 \cong G_2$ are isomorphic groups. Under what conditions on the groups $H_1,H_2$ will we have $$G_1 \times H_1 \cong G_2 \times H_2 ?$$ ...
6
votes
2answers
44 views

Counter example to Mostow's rigidity theorem for 2-manifolds.

I am trying to understand a counter-example to Mostow's rigidity theorem. Here is the counter example I want to understand. Take two non-isometric octagons with the sum of interior angles equal to ...
0
votes
3answers
31 views

Let $G$ be a group and $u \in G$ be a fixed element. By the following, prove that $(G,\bullet)$ is a group.

Let $G$ be a group and $u \in G$ be a fixed element. Define the operation $\bullet$ on G as $\forall a,b \in G, a \bullet b=au^{-1}b.$ Prove that $(G,\bullet)$ is a group. So, I know that in order ...
1
vote
2answers
25 views

Non exact sequence of quotients by torsion subgroups

$$0\rightarrow G_{1}\rightarrow G_{2}\rightarrow G_{3}\rightarrow 0$$ is a short exact sequence of finitely generated abelian groups. We call $\bar{G_{i}}$ the quotient of $G_{i}$ by its torsion ...
0
votes
1answer
24 views

Prove that the group algebras $\mathbb{C}\mathbb{Q}_8$ and $\mathbb{C}\mathbb{D}_4$ are isomorphic. [on hold]

I need to prove that group algebras $\mathbb{C}\mathbb{Q}_8$ and $\mathbb{C}\mathbb{D}_4$ are isomorphic. How can i do this?
0
votes
1answer
56 views

Coproduct of groups

Can anyone explain why the coproduct of groups are the free product? For finite groups, the products are direct products which are also finite. But the free groups are infinite? So the coproduct is ...
2
votes
3answers
41 views

What is wrong with that counting of $S_{3}\times S_{3}$ subgroups?

I want to find all 2-sylow subgroups of $S_{3}\times S_{3}$. I know that there are nine such subgroups, but I tried to count them in the following way - I know that every 2-sylow subgroup isomorphic ...
2
votes
3answers
39 views

Question about group theory and order of elements

Let $G$ be a group and $x, y \in G$. Prove that $ord(x)=ord(y^{-1}xy).$ Let $n,m$ be integers such as $x^n=1$ and $(y^{-1}xy)^m=1$. $x^n=(y^{-1}xy)^m=y^{-1}x^my=1$ I'm not sure how should I ...
2
votes
1answer
23 views

If G is a compact semisimple Lie group and Z is its center, is G/Z always compact?

The title pretty much sums up the question: Suppose $G$ is a compact semisimple Lie group with center $Z$, The question is if $G/Z$ is always compact? or, under which conditions will it be compact?
0
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0answers
16 views

Is $BU^-$ open in GL(n,C)?

Given $G= GL(n, \mathbb{C})$ seen as a Lie Group, let B be the Borel subgroup of upper triangular matrices and $U^-$ be the subgroup of unipotent lower triangular matrices (i.e. lower triangular ...
2
votes
1answer
50 views

Show there is a homomorphism from $G/N$ onto $K$.

Let $\sigma: G \to K$ be an epimorphism (onto homomorphism). And let $N$ be a subgroup of $\ker( )$ that $N \triangleright G$. Show there is a homomorphism from $G/N$ onto $K$. Note that if ...
0
votes
2answers
31 views

Infinite order group that has no nontrivial subgroup?

Is there any infinite order group that has no nontrivial subgroup? I guess there isn't, but I don't know how to approach.
2
votes
1answer
47 views

For which $n\in\mathbb{Z}^+$ is there a $\sigma\in S_{14}$ such that $|\sigma|=n$?

For which $n\in\mathbb{Z}^+$ is there a $\sigma\in S_{14}$ (the group of permutations on $\{1,2,\dots,14\}$) such that $|\sigma|=n$ (where $|\sigma|$ is the order of $\sigma$)? I know you could just ...
2
votes
2answers
23 views

Product of two arbitrary elements in $O(2)$

An exercise in a book I'm reading asks to describe the product of two arbitrary elements in $O(2)$. I would like to solve the exercise but I got stuck. I know that an element in $SO(2)$ can be written ...
1
vote
4answers
235 views

Infinite coproduct of abelian groups

One can see on every text (book, lesson, comments) that a direct sum/coproduct of abelian groups is the same as a finite product but in the infinite case, the direct sum/coproduct is only a subgroup ...
3
votes
2answers
37 views

Finite groups and topological spaces

Can we connect topological spaces with groups as: For topological space $X$ take biective homomorfisms $\phi: X\to X$, then divide such homomorphisms on classes of equivalency $\phi_1 \equiv\phi_2$ ...
4
votes
1answer
52 views

Commutative generators of a group

If a group has commutative generators is the group always abelian? I have a question dealing with how to determine if a Cayley graph of a group is an abelian group. It seems that if the generators ...
0
votes
0answers
7 views

Rewriting a sum of Young Tableaux as Tensors

Is there a straightforward way (perhaps a software) that can write a direct sum of Young Tableaux in terms of tensors? For instance the direct product in $SU(3)$ (taken from this post) ...
1
vote
1answer
30 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 ...
2
votes
1answer
32 views

Isomorphic Coxeter Groups

Is it true that two Coxeter groups having the same Coxeter matrix (equivalently, the same Coxeter graph) isomorphic? Because otherwise, the definition of a Coxeter group from its Coxeter matrix does ...
1
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2answers
34 views

Find the number of homomorphisms from $\mathbb Z_5 $to $S_5$.

Find the number of homomorphisms from $\mathbb Z_5 $to $S_5$. Let $\phi $ be a homomorphism.Then $\dfrac{\mathbb Z_5 }{kerf}\cong Im f$.Now $Im f$ is a subgroup of $S_5$ .Since $kerf $ is a subgroup ...
3
votes
0answers
20 views

Is there a simple group and a subgroup with intermediates lattice L30?

Let $G$ be a finite simple group and $H$ a subgroup. We consider the lattice of intermediate subgroups between $H$ and $G$, noted $\mathcal{L}(H \subset G )$. Let $\mathcal{L}_n = ...
1
vote
1answer
24 views

Inverse of a product in a group can be written as the product of the inverses of each element in reverse order

Let $(G,\circ)$ be a group and let $g_1,...,g_n\in G, n\in\aleph$. Prove that $(g_1\circ ...\circ g_n)^{-1}=g_n^{-1}\circ ...\circ g_1^{-1}$ I tried this by induction but was unsure how to take out ...
1
vote
1answer
32 views

calculate the number of sylow p subgroups of a5

Calculate the number of Sylow $p$-subgroups of $A_5$ We have $|G|=60=2^2\cdot 3\cdot 5$ Let $n_p$ be the number of Sylow $p$-subgroups of $G$. By Sylow's third theorem, we have $n_3\in\{1,4,10\}$. ...
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vote
2answers
32 views

Injective Homomorphism from a group into $GL_n$

$|G|=n\ge 2<\infty,$ A group, I need to know which of the followings are true? $\exists$ allways an injective homomorphism from $G$ into $S_n$ $\exists$ allways an injective homomorphism from $G$ ...
4
votes
1answer
57 views

Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$

Let $G$ be a finite abelian group and let $\phi: G \rightarrow G$ be a group homomorphism. I am trying to show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times ...
5
votes
1answer
75 views

Base for symmetric group

Given symmetric group $S_n$, is it possible to find $k=\lceil\log_2S_n\rceil=\lceil\log_2n!\rceil$ members $\{\alpha_i\}_{i=1}^{k}$ in $S_n$ such that every member of $S_n$ can be written as ...
1
vote
2answers
84 views

Cyclic finite groups and their direct product

Let $C$ be a cyclic group with three elements. prove that $C \times C \times C$ can not be generated by two elements. I was thinking that showing that $C ^{3}$ is isomorphic to ${\mathbb{Z}_{3}} ^{3}$ ...
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0answers
21 views

What does matrix decomposition really mean?

Any element of the symplectic group $\operatorname{Sp}(2n,\mathbb{R})$ can be decomposed using the Euler decomposition into the product of three matrices. \begin{equation} S = O\begin{pmatrix}D & ...
2
votes
1answer
32 views

find about center of G s.t H is normal subgroup of order 2

Let G be a finite group and H be normal subgroup of order 2. Then order of center of G is 0 1 Even integer $\ge $2 Odd integer $\ge $3 I tried this problem by taking G as $S_3$ and H as $ A_3$, ...
3
votes
3answers
71 views

Example where a finite group $G$ of order $n$ has no subgroup of order $m$

Using the Fundamental Theorem of Abelian Groups, one can prove that if $G$ is a finite abelian group of order $n$ such that $m$ is a positive integer that divides $n$, then $G$ contains a subgroup of ...
1
vote
2answers
37 views

Kernel and Image of a group homomorphism

let $G$ be a multiplicative group of non-zero complex analysis.consider the group homomorphism $\phi:G\rightarrow G$ defined by $\phi(z)=z^4$. 1.Identify kernel of $\phi=H$. 2.Identify $G/H$ My ...
2
votes
1answer
33 views

Let R be a ring with unity $1_R$ and let S (with unity $1_S$) be a subring of R. Prove that either $1_S=1_R$ or $1_S$ is a zero divisor of R.

Let R be a ring with unity $1_R$ and let S (with unity $1_S$) be a subring of R. Prove that either $1_S=1_R$ or $1_S$ is a zero divisor of R. My attempt: Let $a \in S$. Then $1_S*a = a$ and this a ...
4
votes
4answers
438 views

Is it always true that $(a,b,c)(a,b,c) = (a,c,b)$?

I noticed that $(1,2,3)(1,2,3) = (1,3,2)$, and I also noticed that $(1,4,3)(1,4,3) = (1,3,4)$. Now, my question is whether or not it is true that for any permutation $(a,b,c)^2 = (a,c,b)$?
2
votes
2answers
31 views

basic question about Group structure (answering a small exercise..)

The operation * defines a binary operation in $\mathbb R\times \mathbb R$ by $(X_1,Y_1)*(X_2,Y_2) = (X_1X_2, Y_1X_2+Y_2)$ defines a group structure (i found out..), but shouldn't we exclude the ...
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0answers
30 views

Number theory and Group theory [on hold]

Can you give me any task which contains Number theory and Group theory?
0
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2answers
59 views

Group $G$ s.t. $x^5y^3=x^8y^5=e$ [on hold]

Let $G$ be group with identity $e$, and $x, y$ be two elements of $G$ satisfying $x^5y^3=x^8y^5=e$. Which of following is true? $x=e$, $y=e$; $x=e$, $y \ne e$; $x \ne e$, $y=e$; $x\ne e$, $y \ne e$. ...
2
votes
0answers
33 views

Set of non fixed points of an automorphism

I am trying to prove the following "For an orbifold chart $ (\tilde{U},G,\phi)$ the set of non fixed point of $ g : \tilde{U} \rightarrow \tilde{U} $ where $ 1 \neq g \ \in G$ is dense in $\tilde ...
1
vote
1answer
40 views

Classifying groups of order 60

I want to solve the following problem from Dummit & Foote's Abstract Algebra text (p. 185, Exercise 14): This exercise classifies the groups of order $60$ (there are thirteen isomorphism ...
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0answers
28 views

Automorphisms of Abelian groups

Let $A$ be a free Abelian group and $N$ a characteristic subgroup of $A$ such that $A/N$ is finite. I also know that $Aut(A/N)$ and $Aut(N)$ are both finite. I have to prove that $Aut(A)$ is finite. ...
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vote
1answer
52 views

The center and centralizer of a group.

If $Z(G)$ denotes the center of the group $G$ and, for $a\in G$, $C(a)$ denotes its centralizer, then show that $a\in Z(G)$ if and only if $C(a)=G$. I got as far to to proving if a is in Z(G) then ...
3
votes
1answer
38 views

Show an $R$-module is a direct limit

This is a scenario I've encountered in my class on $p$-adic L functions. Let $G$ be a profinite group which is the inverse limit of a system $(G_i, f_{ij})$ of discrete finite topological groups. ...
4
votes
1answer
44 views

Sylow subgroups of Symmetric Group

The symmetric group (=permutation group) $S_n$ acts on the set $X_n$ of polynomials in $n$ variables $x_1, x_2, \cdots, x_n$ [with coefficients from $\mathbb{Z}/ \mathbb{Q}/$ or any ring of ...
1
vote
1answer
30 views

Why is it not a sufficient condition to conclude that a is a unity based only on the information that $xa = x$ for all $x$ in $R$?

We have a ring $R$ as follows: Why is it not enough to conclude that $a$ is a unity if $xa = x$ for all $x$ in $R$? Is it because it is by definition that the unity satisfies $ax = xa = x$ for all ...
4
votes
1answer
126 views

Are simple algebraic groups absolutely simple?

Let $k$ be a field. By a simple algebraic group over $k$ I mean an affine group scheme $G$ of finite type over $k$ such that $G$ is connected, non-commutative and every normal closed subgroup of $G$ ...
2
votes
1answer
63 views

Presentation of a group isomorphic to $A_4$

I have a group $G$ defined by $G = \langle x,y,z|x^2 = y^3 = z^3 = xyz \rangle$ and we know that $a$ $=$ $xyz$ belongs to the centre of $G$. But im struggling to show that $\frac{G}{\langle a\rangle} ...
2
votes
0answers
25 views

Etymology of normal extensions and subgroups

According to wikipedia, a normal extension is a splitting field of a family of polynomials, and a normal subgroup is one that is invariant under conjugation. Why are normal extensions and normal ...
1
vote
1answer
12 views

Power of two commuting elements in a group is the binary operation of each of the two elements raised to that power

Let $(G,\ast)$ be a group and let $n\in\aleph$. Prove that if g, h $\in G$ commute, then $(g\ast h)^n$=$g^n\ast h^n$
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votes
3answers
58 views

elements of order $3$ in $A_n$

Let $S_n$ be the permutation group on $n$. We know that $A_n \trianglelefteq S_n$. How many elements $ a \in A_5$ have order three. Is there any formula for finding number of elements in $ S_n $ or ...
3
votes
2answers
46 views

For $\phi$ as a homomorphism not onto show not normal subgroup [duplicate]

I am just a little unclear about what I am supposed to do here. Let $\phi: G $ to $J$ be a homomorphism onto all of $J$. Let $H$ be a normal subgroup of $G$ and let $K=\phi(H)$ be the image of $H$ ...