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

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11 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 \to^{\sigma} 3 \to^{\sigma} 4 \to^{\sigma} 5 \to^{\sigma} 6 \to^{\sigma} 2 ...
6
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1answer
33 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 $$G_1 \times H_1 \cong G_2 \times H_2 ?$$ ...
1
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1answer
25 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
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 ...
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0answers
18 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) ...
3
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0answers
21 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
39 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
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
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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$, ...
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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 ...
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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
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
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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 ...
2
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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 ...
2
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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 ...
1
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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 ...
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1answer
41 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
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 ...
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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 ...
2
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0answers
26 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
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} ...
4
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0answers
29 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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0answers
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
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
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
47 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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34 views

Does the decomposition of a Lie group manifold imply a type of group product?

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
64 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
36 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
41 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
31 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
30 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
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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 ...
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1answer
25 views

Simple homomorphism of groups question [duplicate]

Show that a homomorphism of groups also has the property that $f(a^{-1})=f(a)^{-1}$ for all $a \in G$.
3
votes
1answer
28 views

find a special group

Could anyone show me a virtually-nilpotent ( finitely generated, countable discrete) group $G$ such that $G$ is neither finite-by-nilpotent nor virtually abelian? Thanks! Remarks: 1, By a result ...
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0answers
19 views

Find up to isomorphism all the quotient groups of composition series of a group of order $30$.

I can't seem to understand what I should do here... All I did so far is proving that $G$, (such a group), is not simple. But there are many cases, I can't really tell what $n_2,n_3$ and $n_5$ are, ...
1
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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 ...
2
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2answers
29 views

Finding an order of a coset in $A/B$ where $A$ is a free abelian group and $B$ is a subgroup.

Let $A$ be a free abelian group with basis $x_1,x_2,x_3$ and let $B$ be a subgroup of A generated by $x_1+x_2+4x_3, 2x_1-x_1+2x_3$. In the group $A/B$ find the order of the coset $(x_1+2x_3)+B$. How ...
2
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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?
4
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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 ...
2
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2answers
52 views

How many homomorphisms are there from $\Bbb{Z}_6 \to \Bbb{Z}_{18}$?

I need to determine how many homomorphisms there are from $\Bbb{Z}_6 \to \Bbb{Z}_{18}$. I have never solved that kind of question. I do know that orders are preserved and that some elements can be ...
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49 views

Degrees of irreducible complex characters of alternating groups

What is sum of degrees of the irreducible complex characters of the alternating groups? The background of this question is to calculate the diminsion of a maximal torus of the associated Lie algebra ...
3
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1answer
51 views

Group of order 396 isn't simple

Prove that group of order $396=11\cdot2^2\cdot3^2$ is not simple. $n_{11}$ is $1$ or $12$, so I assumed $n_{11}=12$ and tried to look at the action of the group on $Syl_{11}\left(G\right)$ by ...
2
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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 ...
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0answers
52 views

question regarding group theory proof

Can someone please explain the sentence in red?, how does it follow?
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0answers
28 views

Bigenetic properties of finite group [on hold]

Nilpotency, supersolubility and polycyclicity are bigenetic properties of the class of all finite group. Let be : P is property, X be a class o group. We say that P is a bigenetic property of ...
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0answers
27 views

orbits from group [on hold]

Let g is the group Z2 and let 56 points as follows: w:= [ 7, 8, 15, 27, 42, 89, 95, 121, 125, 134, 139, 150, 167, 5 , 10, 11, 18, 30, 45, 92, 98, 124, 128, 137, 142, 153, 170, 8 , 12, 13, 20, ...