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

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3
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54 views

Tarski monster groups with more than one prime

A Tarski monster group is an infinite, finitely generated group where every proper, non-trivial subgroup is cyclic of order $p$ for a fixed prime $p$. These were shown to exist for "large enough" $p$ ...
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2answers
59 views

The multiplicative group of all the $2^n$-th roots of unity

Consider the multiplicative group $G$ of all the (complex) $2^n$-th roots of unity where $n=0,1,2...$. Which of the following statements are true? Every proper subgroup of $G$ is finite, $G$ has a ...
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2answers
77 views

Category with only one object.

Suppose we have the category with only one object - the group G. Why can we think of the morphisms in this category as of the elements of the group G? I would be very grateful for explanation.
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1answer
27 views

Group Theory $Z_2$ representations

I am trying to understand some group theory. In the notes I am following, I am told: Recall the representations of $\mathcal{Z}_2$: Trivial: $\rho_0(e) = 1$, $\rho_0(a)$ = 1 (i) $\rho_1(e) = 1$, ...
3
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0answers
32 views

If a finite group G is nilpotent then G/F is abelian, where F is the Frattini subgroup of G

Let $G$ be a finite nilpotent group. Consider $F$ the Frattini subgroup of $G$, that is, intersection of all maximal subgroup of $G$. Prove that $G/F$ is abelian. What I am trying that G/F is ...
2
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1answer
39 views

Generators of the group of integers exercise

Let $a,b \in \mathbb Z$. (1) Prove that $\{a,b\}$ is a system of generators of $\mathbb Z$ if and only if $(a,b)=1$, where $(a,b)$ is the greatest common divisor between $a$ and $b$. (2)Show that ...
2
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3answers
55 views

Viewing an abelian group using cayley diagram

I cannot understand this way of viewing whether a group is abelian using cayley's diagram: (from Visual group theory book) What I can't understand is that while checking being abelian we check ...
4
votes
1answer
60 views

hint with an exercise algebra

I'm stuck with the following I hope someone could help me. Let $N$ a normal subgroup of $G$. Show that if $[G:N]=4$, exists a normal subgroup $M$ of $G$ s.t. $[G:M]=2$. My idea: Since $G/N$ has ...
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2answers
139 views

Self studying higher mathematics?

I'm fairly well-versed in calculus but I would like to explore beyond calculus. I have looked into the basics of some topics in higher mathematics such as group theory and abstract algebra and they ...
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2answers
67 views

Does $G$ always have a subgroup isomorphic to $G/N$?

Let $G$ be a group and $N$ a normal subgroup of $G$. Must $G$ contain a subgroup isomorphic to $G/N$? My first guess is no, but by the fundamental theorem of abelian groups it is true for finite ...
7
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2answers
105 views

Uniqueness of the direct product decomposition of finite groups

A group $G$ is indecomposable if: $G = H \times K \Rightarrow \{ H,K \} = \{1, G \}$. Then, a finite group $G$ decomposes into a direct product of indecomposable groups: $G = \prod_i G_i$. ...
2
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1answer
40 views

Smallest possible odd integer that can be the order of a non-Abelian group.

Smallest possible odd integer that can be the order of a non-Abelian group. Attempt: A non abelian group means $Z(G) \subset G$ . Hence, it suffices to find the smallest odd integer $n$ such that ...
2
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1answer
48 views

The smallest composite integer $n$ such that there is a unique group of order $n$?

We need to find the smallest composite integer $n$ such that there is a unique group of order $n$. Attempt: Let us suppose that $n= ab$ is a composite integer where $a,b$ are integers such that ...
2
votes
1answer
26 views

Multiplication between a normal subgroup and an arbitrary subgroup.

Given $G$ a group. $N$ a normal subgroup of $G$, and $H$ an arbitrary subgroup of $G$. Prove that $G=NH$ is a subgroup of $G$. I have to prove that $NH=HN$. But for every $h\in H$ we have that ...
6
votes
4answers
147 views

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$?

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$? I wrote it down in an imprecise way on purpose. The notation above is the linear algebra one: ...
4
votes
3answers
79 views

When a group acts on a metric space isometrically, transitively, and faithfully, is it a closed subspace of the isometry group?

Suppose you have a group $G$ acting on $ (M,d)$ a compact metric space by isometries (meaning $d(gx,gy) = d(x,y)$ for all $x,y \in M$ and all $g \in G$), transitively and faithfully. You can define ...
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2answers
57 views

Semidirect Product variations

Is there two semidirect products of order $16$ with $C_2$ normal subgroup? How many groups of the form $C_4 \times C_2$ : $C_2$ are there ? Is it one expression for two groups ? or more?
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1answer
28 views

Lattice isomorphism theorem

I am currently trying to understand lattice theorem (fourth isomorphism theorem), states that if N is a normal subgroup of a group G, then there exists a bijection from the set of all subgroups A of G ...
5
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2answers
93 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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0answers
34 views

Symbols to represent each distinct symmetry of polyhedra

Is there a pictorial or symbolic way to represent each distinct symmetry of a polyhedron?
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2answers
40 views

In $S_4$, what does the expression for the cyclic group $\langle (13),(1234)\rangle$ mean?

In $S_4$, what does the expression for the cyclic group $\langle (13),(1234)\rangle$ mean? I apologize if this is too basic, but I haven't come across such an expression anywhere in my book. Also, ...
4
votes
1answer
114 views

Reduced Group algebras

Take a finite group and a field of characteristic zero. The group algebra is due to Maschke's theorem semisimple so that its a finite direct sum of matrix algebras over division algebras. I like to ...
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0answers
28 views

Soluble group algebras and centralizers

Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and let $H$ be ...
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0answers
23 views

Sum of degrees of irreducible complex characters for certain groups

The sum of the degrees of the irreducible complex characters (not the square sum which is the group order) is relevant do determine the dimension of a maximal torus in the group algebra. I have ...
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43 views

The Heisenberg group over $\mathbb{Z}/2\mathbb{Z}$

This is inspired by a problem from from Dummit and Foote. It asked me to calculate the order of every element in the Heisenberg group over $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$, which is defined as ...
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votes
2answers
35 views

Question about the symmetric group

How do I prove that if $f\in S_k$ and $f^n=f^m=Id$, then $f^d=Id$ where $d=gcd(n,m)$? I tried writing $f^{dn_1}=f^{dm_1}=f^0$ but this does not lead anywhere. I think I should use that $n_1$ and ...
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1answer
61 views

Verify that two linear representations are equivalent

I've a problem in verifying that two linear representations are equivalent. First of all, I have two permutation representations of the group $G=\langle\alpha ,\beta ,\gamma\rangle$ on the set ...
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votes
2answers
2k views

$|G|>2$ implies $G$ has non trivial automorphism

Well, this is an exercise problem from Herstein which sounds difficult: How does one prove that if $|G|>2$, then $G$ has non-trivial automorphism? The only thing, i know which connects a group ...
2
votes
1answer
216 views

Intersection of Normal Subgroups is Normal in Subgroup but Not Group - Fraleigh p. 143 14.35

Show that if H is a subgroup of a group G, and N is a normal subgroup in G, then $H \cap N$ is normal in H. Show by an example that $H \cap N$ need not be normal in G. I can condone the proof hence ...
4
votes
1answer
76 views

Automorphism $f$ so that $f(x)=x^{-1}$ for half the members of the group: is it an involution?

Let $G$ a finite group. Let $f: G \to G$ an automorphism such that at least half the elements of the group are sent to their inverses, i.e $$\mathrm{card}(\{g \in G|f(g) = g^{-1}\}) \geq ...
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1answer
38 views

For every element of a finite group, there are two distinct exponents that produce the same power

I'm starting my group theory course and we arrived to the following demonstration: Let $G$ be a finite group, then, for every $a\in G$, $O(a) \leq |G|$ where $O(a)$ is the order of the element $a$ ...
0
votes
1answer
226 views

Finding the number of elements of order $2$ in a given group

How many elements of order $2$ are there in the group of order $16$ generated by $a$ and $b$ such that $o(a)=8$ and $o(b)=2$ and $bab^{-1}=a^{-1}$? The basic thing i do not understand is that ...
2
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1answer
42 views

The only group of order $255$ is $\mathbb Z_{255}$ ( Using Sylow and the $N/C$ Theorem)

Let $G$ be a group such that $G =255 = 3.5.17$. Let $H$ be a sylow $17$ sub group of $G$. Then, by Sylow's theorem : the number of sylow $17$ sub groups can be $n=1,18,35,...$ . Since, $n$ should ...
2
votes
1answer
45 views

$(A\cap B)C=AC\cap BC$ in an infinite group

Is there an infinite group $G$ such that for every $A,B,C\subseteq G$, $$1\in A\cap B\cap C~~\to ~~(A\cap B)C=AC\cap BC$$
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1answer
63 views

Image of centralizer under an isomorphism

Suppose we have a group isomorphism $\phi: G\rightarrow K$ between two finite groups and let $H$ a subgroup of $G$. Are there any known facts about the image of the centralizer $C_G(H)$ of $H$ in $G$ ...
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1answer
25 views

Group of conjugation mappings isomorphic to original group

Let $G$ be a group, for every $g \in G$ define $c_g: G \rightarrow G, c_g(h)=ghg^{-1}$. Then $C_G := \{c_g \, | \, g \in G\}$ should be isomorphic to $G$. To prove this I want to show that $\phi: G ...
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2answers
37 views

Let $G$ be a group, and suppose that $M$ and $N$ are normal subgroups of $G$. Prove that the intersection of $M$ and $N$ is a subgroup of $G$.

How do I find out the intersection of $M$ and $N$? I know $M$ and $N$ both contain the identity of $G$. Besides, if I prove the intersection between $M$ and $N$ is a subgroup, how could I prove it is ...
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0answers
31 views

Matrix representation associated to a permutation representation

I have just begun studying group representation theory. I don't understand how I can find the matrix representation associated to a permutation representation. Should I identify each permutation ...
2
votes
1answer
44 views

$S=\{a\in G:f(x)=f(ax)\; \forall x\in G\}$ is a subgroup of $G$

Let $G$ be a group and $f:G\rightarrow G$ a function. Let $S=\{a\in G:f(x)=f(ax)\; \forall x\in G\}$. Prove that $S$ is a subgroup of $G$. This is my first encounter with functions in this ...
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2answers
75 views

Groups with cyclic Commutator subgroup

Is anything known about class of groups with cyclic commutator subgroup?
2
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1answer
771 views

How many mutually non-isomorphic Abelian groups whose order is $3^{2}*11^{4}*7$ are there?

How many mutually non-isomorphic Abelian groups whose order is $3^{2}*11^{4}*7$ are there? Can anyone give a quick (obviously logical) solution?
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1answer
42 views

Is there a concise argument for why this group operation is associative?

Given disjoint groups $(G,\cdot)$ and $(H,\ast)$ and an isomorphism $f:G\to H$, I've been able to show that the binary operation $\diamond$ on $G\cup H$ defined by $a\diamond b = a\cdot b$ if $a,b ...
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0answers
30 views

$H$ is a given subgroup such that for any subgroup $K$ , $HK$ is also a subgroup , then is $H$ normal in $G$? [closed]

Let $H$ be a subgroup of a group $G$ such that for any subgroup $K$ , $HK=KH$ , then is it true that $H$ is normal in $G$ ?
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1answer
59 views

Confusion about Actions of the Symmetric Group

I'm working on some practice questions and I am having trouble understanding actions of the symmetric group. I have the answers, but there were no explanations as to how they were derived. I feel ...
2
votes
1answer
23 views

Show $C^\prime $ is a subgroup of $G$

Let $G$ be a group. Let $C^\prime =\{ a\in G:(ax)^2=(xa)^2 \;\forall x\in G \}$. Prove $C^\prime$ is a subgroup of $G$. I could easily show it for a similar problem but instead with $ax=xa$. I am ...
1
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1answer
106 views

Automorphism of $\mathbb{Q}$

How can we show that any automorphism of $\mathbb{Q}$ under addition is of the form $x \to qx$ ,for some q in $\mathbb{Q}$. edited- I found the same question was asked by ...
2
votes
1answer
30 views

Core of a subgroup and Norm

I have recently faced to a notion called the Norm of a group. What is the relasion between Norm and Core of a subgroup of a finite group? Can we say that for every subgroup, the Core is contained in ...
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4answers
384 views

Does every group have a 'cyclization'?

Here's the question: Does every group have a 'cyclization'? That is, let $G$ be a group. Does there necessarily exist a cyclic group $C$ and a surjective homomorphism $\varphi:G\rightarrow C$ such ...
3
votes
1answer
44 views

Are there uncountably many distinct group operations on an uncountable set?

Call two group operations $\ast_1$ and $\ast_2$ on a set $S$ $distinct$ if there exist $s_1,s_2\in S$ such that $s_1\ast_1 s_2 \neq s_1 \ast_2 s_2$. I know that there are uncountably many distinct ...
6
votes
1answer
70 views

Probability that an element belongs to an infinite subset of a set

$\newcommand{\Sym}{\operatorname{Sym}}$ I was studying $\Sym(\mathbb{N})$, the set consisting of all the bijections from $\mathbb{N}$ to itself. Since it is a group, the concept of "period of an ...