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

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-5
votes
2answers
55 views

Identify quotient group

Identify quotient group $\mathbb R^*/\mathbb R^+$ where $\mathbb R^*$ is multiplicative group of non zero reals and $\mathbb R^+$ denotes subgroup of positive real numbers. I'm using first isomorphism ...
7
votes
3answers
110 views

When are two direct products of groups 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 $$G_1 \times H_1 \cong G_2 \times H_2 ?$$ ...
2
votes
1answer
24 views

Does $G\to X$ have dense image iff $T_eG \to T_{\theta(e)}X$ is surjective?

Let $G$ be a connected algebraic group, $X$ a variety and and $\theta : G\to X$ be a morphism of varieties. (In particular it could be the orbit of some action of $G$.) Consider the corresponding ...
0
votes
2answers
51 views

$(\mathcal{P}(\mathbb{N}),\cap)$ - is a group or not?

I am trying to prove that $(\mathcal{P}(\mathbb{N}),\cap)$ is a group but I'm not really sure my proof is correct. When checking for the identity element I found that for every $A$ in ...
1
vote
0answers
12 views

Proof of a Proposition regarding the reduction of N-torsion groups on elliptic curves

In Diamond-Shurman A first course in Modular forms p.334 Prop. 8.4.4. It is stated, For E elliptic curve over $\bar{\mathbb{Q}}$ with good reduction at the prime ideal $\mathfrak{p}$ the reduction ...
4
votes
1answer
78 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 = ...
0
votes
0answers
13 views

Good reference for The Differntiable Slice Theorem

I am looking for a book that will give me a good proof of The Differentiable Slice Theorem - Suppose a compact Lie group $G$ acts smoothly on a manifold $M$. Then every orbit has a $G$-invarient ...
0
votes
1answer
24 views

Order of normalizer in $S_6$

Find order of normalizer of permutation $s= (12)(34) \in S_6.$ I tried it and I thought we need all permutations $p$ s.t $psp^{-1}=s,$ I wrote down such $p,$ I counted $8.$ But in book answer is ...
1
vote
1answer
16 views

Prove that $P_1\cap P_2$ is $p$-Sylow subgroup of $N_{G}\left(P_1\right)\cap N_{G}\left(P_2\right)$

Let $P_1, P_2$ be $p$-Sylow subgroups of $G$, Show that $P_1\cap P_2$ is $p$-Sylow subgroup of $N_{G}\left(P_1\right)\cap N_{G}\left(P_2\right)$. Don't have any idea.. Thanks !
0
votes
2answers
38 views

Find $o\left (\frac{G}{Z (G)}\right) $ [on hold]

Let $G :=\{a^k, a^k.b|0\le k\lt 9\} $ s..t $o(a)=9$ and $o(b)=2$ and $ba= a^{-1}b.$ If $Z(G)$ denotes center of group $G,$ find the order of $G/Z(G).$ In book answer is $18.$
2
votes
0answers
32 views

Fiber product of non-abelian groups.

I am trying to understand whether surjectivity is needed for a fiber product of non-abelian groups to exist. I seem to have checked that the usual construction works for groups without any ...
3
votes
1answer
41 views

When a normal subgroup $N$ admits a complament?

Let $G$ be a finite group and let $N$ be a normal subgroup. I am looking for conditions on $N$ (and maybe also on $G$) such that there exist a subgroup $H$ of $G$ such that $$G=N\rtimes H.$$ Clearly, ...
0
votes
1answer
21 views

Show that $G$ contains elements $a,b$ s.t. $a^2=b^3=e$ and $ aba=b^2=b^{-1}$

Given that $|G|=6$ and is not commutative. I've showed that generators of $G$ have periods either $2$ or $3$, since $|H_a|=\text{period of generator a}$ divides $|G|$. Since $G$ is not commutative it ...
1
vote
0answers
47 views

Decomposition of a group manifold; is there an associated group decomposition?

The real symplectic group manifold is diffeomorphic to this Cartesian product of manifolds: \begin{equation} \operatorname{Sp}(2n,\mathbb{R}) \simeq \operatorname{U}(n) \times \mathbb{R}^{n(n+1)}. ...
1
vote
0answers
36 views

Is there such a notion of “expansion” in groups?

Given a subset of elements of a finite group $G$, I would like it to be such that the set of all distinct words (as elements of $G$) that can be formed from this set is exponentially large in the size ...
6
votes
2answers
55 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 ...
1
vote
3answers
45 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 ...
1
vote
1answer
38 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
58 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
42 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
25 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
votes
0answers
20 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
252 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
62 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
53 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
vote
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 ...
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
33 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 $n_3\in\{1,4,10\}$. ...
1
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
58 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
76 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 ${\mathbb{Z}_{3}} ^{3}$ ...
1
vote
0answers
22 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
33 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 ...
2
votes
2answers
55 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
39 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
33 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 ...
0
votes
0answers
32 views

Number theory and Group theory [on hold]

Can you give me any task which contains Number theory and Group theory?
0
votes
2answers
61 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? $x=e$, $y=e$; $x=e$, $y \ne e$; $x \ne e$, $y=e$; $x\ne e$, $y \ne e$. ...