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

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$(\mathcal{P}(\mathbb{N}),\cap)$ - 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 ...
2
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0answers
18 views

Order of the Rubik's cube group

Associated to the Rubik's cube is a group as described in this Wikipedia article: $G = \langle F, B, U, L, D, R\rangle$. For example, the element $F$ corresponds to rotating the front face clockwise ...
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1answer
16 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 ...
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6 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 ...
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0answers
12 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 ...
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1answer
16 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
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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 !
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0answers
21 views

A normal intermediate subgroup in L30 lattice?

Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of intermediate subgroups between $H$ and $G$. An intermediate subgroup $H \subset K \subset G$ is a ...
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2answers
34 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 $18.$
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88 views

For all $x,y\in G$ we have: $f(xf(y))=f(x)y$. Prove that $f$ is an isomorphism?

A group $G$ and a function $f:G\longrightarrow G$ are given and for all $x,y\in G$ satisfying $f(xf(y))=f(x)y$. Prove that $f$ is an isomorphism?
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1answer
18 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 ...
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0answers
27 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
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1answer
35 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, ...
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0answers
29 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 ...
2
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0answers
36 views

Find orbit of $1$ for $\sigma$

$\sigma = \left( \begin{array}{cc}1&2&3&4&5&6\\3&1&4&5&6&2\end{array}\right)$ $ 1 \mathop{\rightarrow}^{\sigma} 3 \mathop{\rightarrow}^{\sigma} 4 ...
7
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3answers
103 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 $$G_1 \times H_1 \cong G_2 \times H_2 ?$$ ...
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1answer
35 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?
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3answers
41 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 ...
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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 ...
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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) ...
4
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1answer
55 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 = ...
3
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2answers
54 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 $\phi_1 \equiv\phi_2$ ...
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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
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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$, ...
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2answers
40 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 ...
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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?
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2answers
60 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
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1answer
34 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 ...
2
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0answers
37 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 ...
2
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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 ...
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1answer
31 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 ...
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1answer
42 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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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 ...
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2answers
46 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 ...
2
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0answers
27 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 ...
2
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1answer
74 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} ...
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32 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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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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15 views

How to expand powers of multiple pairwise commuting elements in a group [on hold]

Let (G, $\ast$) be a group and let n $\in\aleph$. Prove that if $g_1,...,g_k\in G, k\in\aleph$ are pairwise commuting elements of G, then $(g_1\ast...\ast g_k)^n$=$g_1^n\ast ...\ast g_k^n$
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1answer
53 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
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1answer
46 views

Can every group be extended to ring with idenity [duplicate]

Can every abelian group converted into ring(by defining multiplication operation) with identity with same order. We can convert every group G into ring by defining a.b = 0 for all a and b in G. But ...
2
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1answer
46 views

Every finite Set as non-abelian Group

For what values of n, we can find a non abelian group. The facts I have proved till now: 1. For n prime there exist only one group upto isomorphism which is cyclic hence abelian 2. For n = 4, there ...
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0answers
48 views

Simplifying a direct sum $\mathbf{3}\oplus\mathbf{3}\oplus\mathbf{2}$ etc

In particle physics, one often uses the dimensionality of the irrep to label the irrep (apparently this is not a very good idea since the dimension does not unambiguously determine the rep.). What are ...
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0answers
46 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)}. ...
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2answers
65 views

How to show that $3\mathbb{Z}/15\mathbb{Z} \cong \mathbb{Z}/5\mathbb{Z}$?

How to show that $3\mathbb{Z}/15\mathbb{Z} \cong \mathbb{Z}/5\mathbb{Z}$ as $\mathbb{Z}$-module over $\mathbb{Z}$? My proof: Define surjective function $f:3\mathbb{Z}/15\mathbb{Z} \rightarrow ...
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0answers
37 views

Inverse property for groups Proof

I was wondering if (1) this proof is correct, and (2) if other proofs exist for the following: Prove that $(a_1a_2...a_n)^{-1}=a_n^{-1}a_{n-1}^{-1}...a_1^{-1}$ where $a_i \in $ a Group $G$ Proof by ...
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0answers
42 views

Under which conditions two groups of order $n=2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13$ are isomorphic

$G,H$ two groups of order $n=2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13$ under which of the following conditions $G$ isomorphic to $H$ (prove or give a counterexample) 1) $G,H$ have have same ...
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0answers
34 views

Prove that all groups of order $3^k5^l$ solvable given $k \le 3$

Prove that $G$ is solvable given its order is $3^k5^l$ while $k,l \in \mathbb{N} , k \le 3$. we are not allowed to use burnside's theorem and Feit–Thompson. I tried to use sylow's theorems to prove ...
1
vote
1answer
31 views

How many possible isomorphisms do we have between G and H? [duplicate]

Let $G=(Z_4,+)$ and let $H=(U_5,*)$ where $U_5 = \{[1],[2],[3],[4] \}$ . I know that $[1]$ and $[3]$ are both generators for $G$. I also know that $[2]$ and $[3]$ are both generators for $H$. In order ...
4
votes
1answer
46 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 ...