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

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Itzykson-Zuber integral over orthogonal groups

I would like to know is there a closed form expression for the following Itzykson-Zuber integral for the orthogonal case. $I=\int_{\mathcal{O}(p)} ...
3
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1answer
48 views

Characterizing finite non-abelian groups in which every subgroup is abelian

How to prove: A non-abelian finite group in which every subgroup is abelian has order divisible by at most two primes.
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2answers
27 views

Let $G$ be a group and $X$ set. $a,b \in X$ are in the same orbit, so show that $stab(a) \cong stab (b)$

Let $G$ be a group and $X$ set. $a,b \in X$ are in the same orbit, so show that $\mathrm{stab}(a) \cong \mathrm{stab}(b)$. What I tried so far: $a,b$ are in the same orbit. So that means $b \in ...
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3answers
41 views

$SL(V)$ and $PSL(V)$ act $k$-transitively on the space of all $1$-dimensional subspaces.

A group $G$ acts $k$-transitive on some set $X$ if for every two $k$-tupels $(x_1, \ldots, x_k)$ and $(y_1, \ldots, y_k)$ there exists some $g \in G$ such that $$ g\cdot x_1 = y_1, \ldots, g\cdot ...
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36 views

Am I doing anything wrong? Find whether the following are group actions

Let $G=\mathbb Z$ be a group and the set is $X=\mathbb Q$. Check whether the following are group actions: $p(z,x)=2^zx$ $p(z,x)=(z+1)x$ $p(z,x)=x^{z+1}$ But it seems like all three don't have the ...
3
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1answer
25 views

Confusion about the hidden subgroup formulation of graph isomorphism

I am going through Quantum factoring, discrete logarithms and the hidden subgroup problem by Richard Jozsa. On page 13, the author discussed the hidden subgroup problem (HSP) formulation of the graph ...
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1answer
57 views

Determining a group $G$ by looking at the number of homomorphisms $H\to G$

I read somewhere that, given a finite group $G$, its structure is completely determined from the knowledge of the values of $|\{H\to G\}|$ (the number of homomorphisms from $H$ to $G$) as $H$ varies ...
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16 views

Non-central proper normal subgroups of unitary groups over fields

Short version: Can someone give an example of an anisotropic Hermitian form over a field such that its corresponding projective unitary group is not simple? Let $F$ be a (commutative, associative, ...
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1answer
45 views

Blocks in permutation group theory (D&F)

I want to solve the following exercise from Dummit & Foote's Abstract Algebra text: Let $G$ be a transitive permutation group on the finite set $A$. $A$ block is a nonempty subset $B$ of $A$ ...
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18 views

central element of order 2 belonging to G'

Is it true that any central element of order 2 that belongs to the commutator subgroup G' of a group G must be equal to the unit of G?
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2answers
38 views

Index and normal subgroups

I want to show the following. For an infinite group G with only two normal subgroups (G and {e}) holds: There does not exist a non-trivial subgroup of G with finite index. I think i should prove ...
3
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1answer
36 views

The index of the Core of a group

I have to prove the following: Let $G$ be a group and $U$ be a subgroup of $G$. Then it holds: If $U$ has finite index, then $\text{Core}_G(U):=\bigcap\limits_{g\in G}gUg^{-1}$ has also finite index. ...
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1answer
39 views

2-Frobenius Groups of order 25920

A group $G$ is called a 2-Frobenius group if $G=ABC$, where $A$ and $AB$ are normal subgroups of $G$, $AB$ is a Frobenius group with kernel $A$ and complement $B$ and $BC$ is a Frobenius group with ...
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1answer
20 views

Conjugating a permutation

I am trying to see that two permutations are conjugate exactly when they have the same cycle decomposition. I fail to see that $$r(i_1,i_2,\dots,i_k)r^{−1}=(r(i_1),r(i_2),\dots,r(i_k))$$ Any thoughts ...
2
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2answers
80 views

What is $\operatorname{Aut}(\mathbb Z_4\times\mathbb Z_2)$?

I'm trying to figure out $\operatorname{Aut}(\mathbb Z_4\times\mathbb Z_2)$, but I'm not really sure how to go about this. I feel like there are some non-obvious generators.
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53 views

Orbits of the group action $g. (x,y) = (gx,gy)$ on cartesian product

Let $G$ be a group acting on a set $X$. Then we have a natural action on $X \times X$ in the following way: $g. (x,y) = (gx,gy)$. Then, suppose we have two points of interest $x_1,x_2 \in X$, and we ...
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the homophonic group: a mathematical diversion

By definition, English words have the same pronunciation if their phonetic spellings in the dictionary are the same. The homophonic group H is generated by the letters of the alphabet, subject ...
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1answer
115 views

What is the manifold structure of U(n)?

A Lie group is simultaneously a differentiable manifold. As I understand it, the Lie group is generated via exponentiation of the generators of the Lie algebra. It is intuitively clear to me that the ...
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0answers
35 views

Infinite finitely generated $2$-groups

Can somebody give me an example of a finitely generated infinite $2$-group?
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1answer
40 views

Finite Group with Nilpotent Subgroup of Prime Power Index is Solvable

Let $G$ be a finite group, and assume that $H$ is a nilpotent subgroup whose index is a prime power. WLOG, we can say that the index of $H$ is the highest power of $p$ which divides the order of $G$. ...
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1answer
26 views

Are augmented algebra maps of group algebras group homomorphisms?

Given a finite group $G$ and a field $k$, we can form the group algebra $kG$ with basis the elements of $G$. There is a natural augmentation $\varepsilon\colon G\to k$ that sends an element to the sum ...
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1answer
13 views

Group of permutations and disjoint cycles.

Two cycle $[i_1,..., i_r]$ and $[j_1,...,j_s]$ are said disjoint if no integer $i_\eta$ is equal to any integer $j_{\mu}$. Prove that a pemutation is equal to a product of disjoint cycles. I was ...
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4answers
113 views

abstract Algebra (group theory)

Three coins are placed on a table; showing heads. Can you get all the coins to show tails, by turning over two coins at a time? Use Group Theory to prove your answer. I know that the answer is no I ...
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2answers
38 views

Show $D_3\cong S_3$ and $D_n\ncong S_n$ for $n\gt 3$

Show that $D_3\cong S_3$ and $D_3\ncong S_3$ for $n\gt 3$, where $D_3$ denotes the dihedral group and $S_3$ the symmetric group. I define a group isomorphism between $D_3$ and $S_3$. Both group ...
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2answers
79 views

Explicit description for $G=\langle a,b,c\mid[a,b]=b\,,\,[b,c]=c\,,\,[c,a]=a\rangle$

I am trying to give an explicit description of the group $$G=\langle a,b,c\mid[a,b]=b\,,\,[b,c]=c\,,\,[c,a]=a\rangle\,.$$ Generalizing to fewer generators, one ends up with the trivial group, i.e. ...
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1answer
109 views

Subset of $\mathbb{Z} \times \mathbb{Z}$

I have a past exam question that is as follows: Let $k$ be a fixed integer and $S = \{(a,ka)|a \in \mathbb{Z}\}$ be a subset of $\mathbb{Z} \times \mathbb{Z}$. Prove that $S$ is a subgroup of ...
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3answers
96 views

Examples of Infinite Simple Groups

I would like a list of infinite simple groups. I am only aware of $A_\infty$. Any example is welcome, but I'm particularly interested in examples of infinite fields and values of $n$ such that ...
2
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0answers
37 views

Counter example of monotone union [duplicate]

I saw this exercise in "Elements of Abstract and Linear Algebra" by E. H. Connell: Suppose $G$ is a group. Suppose $T$ is an index set and for each $t \in T$, $H_t$ is a subgroup of $G$. Furthermore, ...
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129 views

Infinite groups such that $G/G'$ has odd order.

Can someone give examples of an infinite group $G$ such that $G/G'$ has odd order.
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3answers
75 views

How to show that no. of elements $x$ of group $G$ such that $x^3=e$ is odd?

Let G be a finite group G. Then How can I show that no. of elements $x$ of group $G$ such that $x^3=e$ is odd ? I read this question in an Algebra book. Since $e^3=e$, e must be one of those elements. ...
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1answer
62 views

How to prove “a group $G$ of order $72$ can't be a simple group”?

By using Sylow theorem, I can prove that $G$ has either $1$ Sylow $3$-subgroup or $4$ Sylow $3$-subgroup, but I don't know how to continue the proof.
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1answer
128 views

What can we say about groups $G$ with $H_3(G)=0$?

Let $G$ be a group. What can we say about groups such that $H_3(G)=0$? If a characterization is not possible, then knowing examples of such groups would be good? Any help is appreciated. Thanks
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1answer
97 views

examples of polyclic groups

From the notes Coarse differentiation and the geometry of polycyclic groups, I found a theorem $\Gamma$ is polycyclic iff $\Gamma$ is a lattice in a solvable unimodular lie group $G$ - Mostow ...
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1answer
77 views

Finite Subgroups of $GL_2(\mathbb Q)$

I want to prove that the only finite subgroups of $GL_2(\mathbb Q)$ are $C_1, C_2, C_3, C_4, C_6, V_4, D_6, D_8,$ and $D_{12}$. First, we determine all possible finite orders of elements. Now, an ...
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3answers
48 views

Show that: $\frac{D_n}{\langle a\rangle}\simeq\mathbb{Z_2}$

Show that: $$\frac{D_n}{\langle a\rangle}\simeq\mathbb{Z_2}$$ where $D_n$ is dihedral group and $a$ is generator of order $n$.
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59 views

Having trouble understanding what ker f is

I am trying to understand what exactly 'ker f' is. My guess is that it is a set of all of the elements that were lost in the process of a mapping. For example: $f:A\to B$ $f = \left\{ \begin{align} ...
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3answers
68 views

Monotone Union of subgroups being subgroup

I saw this exercise in a book: Suppose $G$ is a group. Suppose $T$ is an index set and for each $t \in T$, $H_t$ is a subgroup of $G$. Furthermore, if $\{H_t\}$ is a monotonic collection, show that ...
3
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1answer
49 views

It is true that every group that has a finite number of subgroups is finite? [duplicate]

It is true that every group that has a finite number of subgroups is finite? I think not, but I can not find counterexamples.
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109 views

Is there any simple proof for this?

Let $G$ be a group with $|G:Z(G)|=n$ then $\phi(x)=x^n$ is a homomorphism from $G$ to $Z(G)$. I guess it has a proof using transfer theory, I wonder whether it has an elemantary proof or not. ...
3
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1answer
55 views

Are all Sylow 2-subgroups in $S_4$ isomorphic to $D_4$?

I was assigned to show that every Sylow 2-subgroups in $S_4$ is isomorphic to $D_4$. So I figured, since $|S_4|=24=2^3\cdot 3$, every Sylow 2-subgroup has either the form: $\langle ...
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1answer
30 views

Understanding the elements in groups(modulo, cyclic and other(?))

Question exactly as given on past exam: Consider the group $G=(\mathbb{Z}/15\mathbb{Z})^\times$ (under multiplication). Let $H$ be the subgroup generated by $2$ (that is, $H = \langle 2 \rangle$). ...
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75 views

A question on Groups and its center

If $G$ be a group of order $8$ and $o(x)=4$ then how to prove that $x^2 \in Z(G)$ ? I can only figure out that $x^2=x^{-2}$ ; Please help
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Monotonic Union of Submodule

The theorem is: Suppose $M$ is an $R$-module, $T$ is an index set, and for each $t \in T$, $N_t$ is a submodule of $M$. If $\{N_t\}$ is a monotonic collection, $\cup_{t\in T} N_t$ is a submodule. Is ...
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1answer
38 views

Order of some $\alpha$ in $S_4$

I want to work out the order of $\alpha = \left[ \begin{align} & 1&2&&3&&4&\\&4&2&&1&&3& \end{align} \right]$ in $S_4$ Now when I think of ...
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1answer
61 views

on automorphisms groups a finite 2-group

Let $G=\langle a,b\mid a^4=b^4=1, bab^{-1}=a^3\rangle$. Please prove that $Aut(G)$ is generated by the automorphisms $$a\mapsto ab,\hspace{10pt} a\mapsto a^3,\hspace{10pt} a\mapsto ab^2, ...
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1answer
44 views

Bounded almost-homomorphisms on the integers

Let $f : \mathbb{Z} \to \mathbb{Z}$ be an "almost-homomorphism": The set $\{f(n+m)-f(n)-f(m) : n,m \in \mathbb{Z}\}$ is bounded. We may assume that $f$ is odd, i.e. $f(-n)=-f(n)$ for all $n$. Assume ...
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1answer
39 views

Isomorphism between groups of $2 \times 2$ matrices

I'm stuck on this problem: For $\mu \in \mathbb{R} \setminus \{1\}$ let $$G_\mu := \left\{\begin{pmatrix}a & b \\ 0 & a^\mu \end{pmatrix} : a \in \mathbb{R}^+, \; b \in \mathbb{R}\right\} .$$ ...
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4answers
108 views

Mapping from $\{1,\ldots,n!\}$ to the symmetric group $S_n$

Is there an easy known bijective mapping formula between the set $\{1,\ldots,n!\}$ and the symmetric group $S_n$? I want to pick a number $k \in \{1,\ldots ,n!\}$ and assign a unique permutation of ...
5
votes
1answer
61 views

Recovering a group action from sizes of orbits of individual elements

Let $G$ be a group (say, finite) and let it act on a set $X$ (say, also finite). For every element $g \in G$, we can consider its action on $X$. My rather vague question is What information about ...
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30 views

Choosing a canonical fundamental domain

I have a set of equations that partitions a certain space into equivalent regions. For a given point $p$ contained in region $R_1$, there are equivalence relations giving its equivalent position in ...