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

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Are there at least denumerably many distinct group operations on any denumerable set?

I believe I've arrived at a messy, complicated proof of the following statement: For any denumerable set D, there exist at least denumerably many distinct group operations on D. I'd prefer to keep my ...
3
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3answers
59 views

An epimorphism from $\mathbb Z⊕\mathbb Z⊕\cdots$ to $\mathbb Q$

I want an explicit example of an epimorphism from $\mathbb Z⊕\mathbb Z⊕\cdots$ to $\mathbb Q$. Thanks.
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1answer
36 views

Showing that the Binary Icosahedral Group (given by a presentation) has order $120$.

Say we have a group generated by $a, b$, with the relations $(ab)^2=a^3=b^5$ (note that these are not necessarily equal to the identity). How do I show that the group has $120$ elements? Without ...
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3answers
46 views

Group of even order must contain $a:a=a^{-1}$ $ (a\not = e)$ [duplicate]

Let $G$ be a finite group. If the order of $G$ is even, prove that there is at least one element $a$ in $G$ such that $a\not= e$ and $a=a^{-1}$. Here's my idea: Suppose $\{x_1,\cdots,x_n\}$ is ...
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1answer
74 views
+50

Whether two quotients of $\mathbb{Z}^2$ are isomorphic.

Let $H_1$ be the subgroup of $\mathbb{Z}^2$ generated by $\{(1,2),(4,1)\}$, let $H_2$ be the subgroup of $\mathbb{Z}^2$ generated by $\{(3,2),(1,3)\}$. Is it true that $\mathbb{Z}^2/H_1\cong ...
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2answers
50 views

If $p\mid|G|$ then how many elements of order $p$ are there in $G$?

Let $G$ be a finite group and $p$ be a prime such that $p\mid|G|$ , then obviously $G$ has an element of order $p$ (by Cauchy's theorem) ; I would like to know exactly how many elements of order $p$ ...
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3answers
147 views

Isomorphism of $\mathbb{Z}/ \mathbb{nZ}$ and $\mathbb{Z}_n$

Let $n\in \mathbb{Z}^+$. How do I prove that $\mathbb{Z}/n\mathbb{Z}$ is isomorphic to $\mathbb{Z_n}$? Is there any good homomorphism $\phi$ I could use that graphs $\mathbb{Z}/n\mathbb{Z}$ to ...
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2answers
47 views

Show $H_n$ is a subgroup of $G$ for any $n\in\mathbb N$

Let $G = GL_2(\mathbb C)$ be the group of invertible complex $2 \times 2$ matrices and for each $n\in\mathbb N$ consider the subset: $$H_n = \{\,A \in G : (\det A)^n= 1\,\}.$$ I know to prove if it ...
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1answer
67 views

In a group of exponent $2^n$, $[x^{2^{n-1}},y^{2^{n-1}},\ldots,y^{2^{n-1}}]=1$?

In a group of exponent $2^n$, is the following equality true? $[x^{2^{n-1}},\underbrace{y^{2^{n-1}},\ldots,y^{2^{n-1}}}_n]=1$. Here, $[a, b, c]=[[a, b], c]$. Call the above question "Question ...
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6answers
51 views

Generators of the Symmetric group $S_3$

I am trying to find the generator(s) of the Symmetric Group $S_3$ and I have attempted this via brute force by listing the permutations of $S_3$ and composing and repeating them but I have not found ...
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2answers
86 views

Properties of an element $x\in X$ in the Cayley-Graph $\Gamma(G,X)$ of a group G.

My problem is the following: Let $G$ be a group with generating set $X$. We can look the Cayley-Graph $\Gamma(G,X)$ of $G$. Let $x\in G$. Then it holds: $d_{\Gamma}(v,xv)\leq 1$ for all $v\in ...
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1answer
176 views

proof that finite group of rotations of plane is cyclic

How can we prove that any finite group of orientation-preserving isometries of the Euclidean plane is cyclic, using the following hint? 'You may assume that given any non-empty finite set E in the ...
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425 views

How strong is the statement that Thompson F is amenable?

Justin Moore's proof turned out to have an error I just attended Justin Moore's talk on this today. Since I am neither a group theorist nor a combinatorist, and is not familiar with ultrafilters I ...
3
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1answer
114 views

Virtually infinite cyclic groups act on a tree

A virtually infinite cyclic group $G$ is quasi-isometric to $\mathbb{Z}$ and thus has two ends; by Stallings theorem, $G$ acts (without inversion) on a tree with finite edge-stabilizers. But the ...
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2answers
128 views

Extensions and Kazhdan's Property (T)

Is Kazhdan's property (T) stable under extensions? i.e. if $G$ is an extension of a group with property (T) by a group with property (T), does it follow that $G$ has property (T)?
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1answer
18 views

Question about particular words in the free group on three generators

In the free group generated by the letters $x,y,z$ suppose that we have a word such that for any one of $x,y,z$ the indices of each occurrence of that letter in our word sum to zero. Suppose further ...
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2answers
208 views

The index of a subgroup is divisible by the index of its image under a homomorphism

Let $f$ be a homomorphism defined on a finite group $G$, and let $H$ is the subgroup of $G$. Then show that $$ \left [f(G) : f(H)\right] \text{ divides } \left [G : H\right].$$ I know $$\left [G : ...
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1answer
49 views

Constructing a group with given normal subgroups.

Let $N_1,N_2,\dots,N_n$ be simple groups. Is there is a group $G$ with exactly $n$ nontrivial proper normal subgroups isomorphic to $N_1,N_2,\dots,N_n$?
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1answer
60 views

Proving Finiteness of Group from Presentation

Given the group $G = \langle a, b, c : a^2 = b^3 = c^5 = abc\rangle$, I want to show that $H = G / \langle abc\rangle$ is a finite group. I tried to find a canonical form for elements of $H$. That ...
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1answer
77 views

How does $A_n$ look in Aut$(X)$?

Let me phrase my question precisely: Let $X=\{1,2,3,...,n\}$, $ \ S_n=\mbox{Sym}\{1,2,3,...,n\}$ be symmetric group on $n-$letters. Let $\mbox{Aut}(X)$ denote the automorphism group of $X$. We ...
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1answer
25 views

A Group Isomorphic To The Direct Product Of Two of its Subgroups. Then are These Subgroups Normal?

Let $G_1$ and $G_2$ be subgroups of a group $G$. Assume that $G$ is isomorphic to $G_1\times G_2$. Then is it necessary that $G_1$ and $G_2$ are normal in $G$? Clealry $G_1\cong ...
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1answer
69 views

Computation of the cokernel of the map $f: \mathbb Z \rightarrow \mathbb Z_{(2)} \oplus \mathbb Z $ defined by $f(1)=(1,2)$

$f: \mathbb Z \rightarrow \mathbb Z_{(2)} \oplus \mathbb Z $ (the sum is direct) $f(1)=(1,2)$ so the image is $\mathbb Z*(e_{1}+2e_{2})$ however computing the cokernel of this map really puzzles me ...
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1answer
488 views

Exercises in Group Cohomology

I'm interested in finding a textbook to learn group cohomology, a book that contains a lot of examples and also a lot of good exercises to test my understanding. I would appreciate some feedback. ...
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1answer
198 views

Maximal Subgroups and order of a group

I encountered the following exercise in Isaacs' Algebra: "Suppose a group $G$ has only one maximal subgroup. Prove that the order of $G$ must be a power of a prime". I think I've proven this for ...
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1answer
84 views

What does $\frac12(D_{2p}\times D_{2p})$ mean in group theory?

Reading a thesis, I have come across the (unexplained) notation $$\frac{1}{2}(D_{2p}\times D_{2p})\cong (p\times p):2,$$ where $D_{2p}$ is a dihedral group. What does this "$\frac12$" notation mean? ...
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0answers
46 views

Variational formulations in group theory?

I apologise if this is a naïve question. Are there any known / widely applicable / important variational formulations in (finite) group theory? That is, a relationship of the form $$\alpha(G) = ...
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1answer
20 views

Explicit formula for invariant inner product of the standard representation of $S_3$

Let $V$ be a representation of a group $G$ over $\mathbb{C}$. Given the standard Hermmitian inner product $\langle\cdot,\cdot\rangle$ on $V$ we can always define a $G$-invariant inner product by ...
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1answer
181 views

Is true that : group of exponent 4 implies that $[[[x,y],y],y]= \text{identity}$?

It is well known that: If the square of every element of a group is the identity then the group is abelian. Also is known that: In a group, if (for all $x$) the cube of $x$ is the identity (i.e. a ...
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1answer
21 views

Finite Linearly Ordered Abelian Monoids

This question concerns the proper definition of the phrase "finite linearly ordered abelian monoids". The sequence A030453 of OEIS counts the number of "finite linearly ordered abelian monoids". The ...
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49 views

On the definition of groups of multiplicative type

Let $k$ be a field of characteristic 0. The definition of a linear algebraic $k$-group of multiplicative type (m.t.) I've seen the most in the literature is that $G$ is of m.t. if it is a ...
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4answers
138 views

Normal Groups and Quotient Groups

These concepts are currently confusing me. My reading first defined a normal subgroup as one that is the kernel of a group homomorphism. Then it introduced the terms "left coset" and "right coset," ...
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2answers
30 views

Normal subgroups, direct product and monomorphism problem

Let $G$ be a group and let$H,K$ be normal subgroups of $G$. Let $\pi_H,\pi_K$ be the projections on $H$ and $K$ respectively. Show that the map $$f:G/(H \cap K) \to G/H \times G/K$$ defined as ...
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2answers
20 views

Isomorphism between quotient groups

Exercise Let $f:G \to G'$ be an isomorphism and let $H\unlhd G$. If $H'=f(H)$, prove that $G/H \cong G'/H'$. As I've shown that $H'\unlhd G'$, I thought of defining $$\phi(Ha)=H'f(a)$$I was trying ...
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21 views

Counting of edges coloring in a graph

The problem is to count of coloring graphs. We have three colors. And I found all automorphisms. It is: $$\alpha_1: (1)(2)(3)(4)(5)(6)$$ $$\alpha_2: (123456) $$ $$\alpha_3: (135)(246) $$ ...
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1answer
69 views

How do we know the classification of finite simple groups is finished?

I don't have much experience in group theory beyond knowing what a finite simple group is, as well as the other basic parts of the topic, so far as I can tell, so if the actual answer is too complex ...
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0answers
38 views

For $G$ a group of order $1925$, find the number of Sylow $5$-subgroups

Let $G$ be a finite group of order $1925$. Find the number of Sylow $5$-subgroups in $G$. There must be $1$ or $11$ such subgroups. What is the actual number?
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48 views

No. of homomorphisms from $\mathbb Z_n$ to $\mathbb Q$

How many homomorphisms are there from $\mathbb Z_n$ to $\mathbb Q$ ?
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1answer
21 views

Elements in the same coset and the Cayley Diagram

A question from Visual group theory, by Nathan Carter. In a Cayley diagram, if $aH$ is a coset of a subgroup $H$ of a group $G$ and $b$ belongs to $aH$, why is it that every node that can be reached ...
4
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1answer
51 views

Order of group $GL_{2}\left( \mathbb{F}_{p}\right) $

I'm having a hard time counting. I need to count the number of elements for the multiplicative group of invertible $2\times 2$ matrices $GL_{2}\left( \mathbb{F}_{p}\right) $ with elements from the ...
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1answer
55 views

To show that a concretely defined group is isomorphic to an explicitly presented group, what strategies are available?

I have a homework problem of the following form. We're given presentation of a group $\langle x,y \mid R\rangle$ explicitly, and two matrices $X,Y \in \mathrm{GL}(\mathbb{C},2).$ We know $X$ and $Y$ ...
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2answers
36 views

The number of distinct conjugates of a $p$-subgroup

Let $G$ be a $p$-group and $H$ be a $p$-subgroup of $G$; $H$ is not normal in $G$. Then prove that the number of distinct conjugates of $H$ in $G$ divides $|G|$. Notice that the conjugates may have ...
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1answer
47 views

Constructing a group automorphism making a diagram commute

Let $A,B$ be groups, suppose we have an epimorphism $p:A \to B$. Let $\phi \in \operatorname{Aut}(A)$. Does there exist some $\varphi \in \operatorname{Aut}(B)$ such that the following diagram ...
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9answers
5k views

Example of infinite groups, such that all its elements are of finite order

I am in need of: Example of infinite groups, such that all its elements are of finite order
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1answer
43 views

How to determine the isomorphism types of given groups with generators and relations

I was classifying the all groups of order 30 and I got the following groups $\langle a,b \mid b^{-1}ab=a^4, a^{15}=b^{2}=1\rangle$ and $\langle a,b \mid b^{-1}ab=a^{11}, a^{15}=b^{2}=1\rangle$. How ...
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50 views

Let $G$ be a group and $K$ be subgroup of order $p^a$. I am trying to show that $|\{H \leq G : K \subset H, |H|=p^{a+b}\}| \equiv 1 \pmod{p}$

Let $G$ be a group and $K$ be subgroup of order $p^a$. I am trying to show that $$|\{H \leq G : K \subset H, |H|=p^{a+b}\}| \equiv 1 \pmod{p}$$ for each b fixed & for all $a+b$ such that ...
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1answer
34 views

Irreducibility of a module of a cyclic group

Let $G = \langle x \rangle$ be a cyclic group of order $p$ ($p$ is a prime). Let $M$ be a vector space over $\mathbb Q$ with basis $\{m_0, m_1, \cdots, m_{p-1}\}$. Define $\rho(x)$ to be $$\rho(x)m_i ...
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3answers
132 views

Number of Automorphisms on group

I had a doubt in the following question How many automorphisms possible on the group $Z^{+}_{12} $. Although I believe it should be $12!$ but want to confirm. Thanks
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2answers
178 views

Show that the kernel of the map $SL(n, \mathbb{Z}) \to SL(n, \mathbb{Z}/3\mathbb{Z})$ has no torsion.

I am trying to show that the kernel of the natural map $SL(n, \mathbb{Z}) \to SL(n, \mathbb{Z}/3\mathbb{Z})$ has no torsion. That is, if $A$ is in the kernel then $A = I$ or $A^n \neq I$ for all $n ...
3
votes
1answer
30 views

Problem on normal subgroups

Problem Let $G$ be a group and $H,K$ subgroups of $G$, we define $HK=\{h.k : h \in H, k \in K\}$. Prove that if $H$ or $K$ is normal, then $HK$ is a subgroup. In order to show $HK$ is a subgroup, ...
3
votes
2answers
70 views

Characterize the natural numbers $n$ such that there is a surjective group homomorphism from $S_n$ to $S_{n-1}$.

This has always been one of my most favorite exercises from Group Theory, and I was surprised to see that this hasn't been asked before. To repeat: Characterize the natural numbers $n$ such that ...