A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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Intuitive argument for the transitivity of $PGL(n+1)$ acting on $\mathbb{P}^n$

In spherical geometry we can consider the action of $\operatorname{O}(n+1)$ on the unit sphere $\mathbb{S}^n$. It's easy to see that this action is transitive, because for any two points $x,y \in ...
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6
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139 views

Do uncountable groups have generating sets where you can find elements of arbitrarily long length, and what are conditions to guarantee that?

I would like to find a way to pick a set of generators in a group $G$ so that one can always find an element of $G$ of arbitrary length. I'm not sure whether or not this is always possible, and if ...
3
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4answers
361 views

Does group inverse commute with multiplication for all groups? [duplicate]

Is the following a property of $\textbf{all}$ groups? $a^{-1} \circ b^{-1} = (a \circ b)^{-1}$ As far as I can tell it is true for addition and multiplication, but in the notes that I have come ...
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360 views

Proof of Wilson's Theorem using concept of group.

I am studying group theory so I do it by using the concept of group. What I am trying to prove is if p is prime then $(p-1)!\equiv-1\mod p$ Note that $\mathbb{Z_p}$ forms a multiplicative group. ...
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1answer
27 views

Homomorphic image if smaller fails to exist

Suppose a finite group $G$ has no homomorphic image of order $n$. Is it possible for $G$ to have a homomorphic image of order a multiple of $n$? My gut says "no", as the larger homomorphic image ...
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43 views

Computing Factor Group step-by-step manually

I am reading Fraleigh's Abstract Algebra $\S$15 on factor group, Example #15.11: Compute factor group $(\mathbb Z_4 \times \mathbb Z_6) / \langle (2, 3)\rangle.$ The text gives a solution that ...
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27 views

$x,y \in G$, $o(x) = 2, o(y) = 2, o(xy) = k, k\geq 3$. Show that $D_k \cong \langle x,y\rangle$

I know that $D_k = \{1,r, \dots, r^{k-1}, s, sr, \dots, sr^{k-1} \}$ and that I can use $\phi: s^i r^j \mapsto x^i(xy)^j$. I don't know what can be found using this.
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3answers
75 views

Does the group $\mathbb{R}^{\times} / \mathbb{Q}^{\times}$ have a subgroup of order 5?

Does the group $\mathbb{R}^{\times} / \mathbb{Q}^{\times}$ have a subgroup of order 5? I don't know how I should approach this problem. Could you give me some hints on how to solve this? A more ...
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15 views

Branching rules without previous knowledge of the projection matrix?

Given a representation $R$ of some group $G$ one can find in many books and papers (e.g. page 96ff here) the decomposition under certain subgroups: $$ R= R_1 + R_2 + \ldots$$ This is often called a ...
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2answers
112 views

More Symmetric than the symmetric groups?

So I was considering the following question. Is there a group of size n! (or less), that contains a larger number of subgroups than $$S_n$$? In thinking about this problem one obvious contender that ...
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1answer
18 views

Set of discrete orbits under subgroup of $Isom(\mathbb{E}^n)$ is clopen

I'm trying to solve the following exercise (exercise 1.4 from Szczepanski's "Geometry of Crystallographic Groups"): Let $\Gamma$ be a subgroup of $I(\mathbb{E}^n)$, the group of isometries on ...
3
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1answer
47 views

$M= \{ A \in Mat_{2 \times 2}{\mathbb{R}}| \det(A)=1 \}$ is homeomorphic to $S^{1} \times \mathbb{R}^{2}$

Let's consider a group $M$ (under multiplication) of all matrices $A$ of size $2 \times 2$ over $\mathbb{R}$ so that $\det(A)=1$. How to show that the group is homeomorphic to the $S^{1} \times ...
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1answer
66 views

Terminological conventions regarding group actions

Suppose: $G$ is a group $T$ is a Lawvere theory ("algebraic theory") and $X$ is a $T$-algebra What conventions surround the phrase: "action of $G$ on $X$"? In particular, does this mean: a ...
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1answer
64 views

Matrix Algebras: Generator

Problem Given the algebra $\mathcal{M}_\mathbb{C}(2)$. Denote the normals: $$\mathcal{N}:=\{N\in\mathcal{M}_\mathbb{C}(2):N^*N=NN^*\}$$ And their calculus: ...
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54 views

Is there a simple proof of Frobenius's theorem using Sylow theory? [closed]

I ask because the proof I always stumble upon uses double induction and properties of the totient function. The statement of the theorem is: If $n$ divides the order of a finite group $G$, then ...
3
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1answer
173 views

False proof that simple groups are cyclic

"Proof" of not cyclic $\implies$ not simple: If $G$ is not cyclic, $\langle g \rangle$ is an abelian, and hence normal subgroup of $G$. Doesn't this show every simple group is cyclic? What am I ...
4
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2answers
89 views

Coproducts in $\mathsf{Grp}$

The limits and colimits in the category of abelian groups are as nice as can be, since products and equalizers are the same as in the category of sets. In the category of groups, however, the ...
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2answers
36 views

The intersection of all abnormal maximal subgroups of a given group

Let $G$ be a group and let $\mathrm{aFrat}(G)$ denote the intersection of all abnormal maximal subgroups of $G.$ The English summary of the paper "The intersection of abnormal maximal subgroups of ...
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2answers
36 views

Isometric group of hyperbolic 3-dim manifolds

In the book : Foundation of Hyperbolic Manifolds There is a theorem that any finite subgroup of $ Isom(\mathbb{E}^n) $ fixes a point. And I hope to solve the following question : Any ...
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1answer
43 views

a question of group theory [duplicate]

let $S$ be a collection of (isomorphism classes of) group $G$ which have the property that every element of $G$ commutes only with the identity element and itself then which option is true and why ? ...
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2answers
150 views

Group Theory: group under the composition multiplication modulo $p$

Suppose you have a group $G(S,*)$ where $S=\{1,2,\ldots,p-1\}$, $p$ is prime number, and $*$ is equivalent to the multiplication$\mod p$. If $a,b$ belong to $S$, then $ab\pmod{p}$ also belongs to ...
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$ G $ is supersoluble if and only if every normal subgroup of $ G $ with index no more than 2 satisfies the maximal permutizer condition.

Permutizer of a subgroup $ H $ of $ G $ is defined to be the subgroup generated by all cyclic subgroups of $ G $ that permute with $ H $, i.e. $ \langle x\in G \vert \langle x \rangle H = H \langle ...
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1answer
24 views

Prove that if $H$ is a unique subgroup of order $|H|$ then $H$ is a characteristic subgroup.

Let $G$ be a group and $H\leq G$. $H$ is a unique subrgroup of order $|H|$. Prove that $H$ is a characteristic subgroup. I tried to show by contradiction that if $\phi (h)\not \in H$ then I can find ...
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51 views

condition for a group to be abelian [duplicate]

I’d like to prove: Let $G$ be a group and $m,n$ two relatively prime numbers. If $x^my^m=y^mx^m$ and $x^ny^n=y^nx^n$ for all $x,y\in G$, then $G$ is abelian. Thanks for help in advance.
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$K/F$ Galois Extension, $H \leq G$, where $G$ the Galois group, then there is $\alpha \in K$ s.t. $H=\{\sigma \in G : \sigma \alpha = \alpha\}$.

Let $K/F$ be a finite Galois extension of fields with Galois group $G$. Let $H$ be a subgroup of $G$. Then there is $\alpha \in K$ such that $H=\{\sigma \in G : \sigma \alpha = \alpha\}$. My proof ...
4
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3answers
130 views

Every normal subgroup is the kernel of some homomorphism

Clearly the kernel of a group homomorphism is normal, but I often hear my professor mention that any normal subgroup is the kernel of some homomorphism. This feels correct but isn't entirely obvious ...
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Subgroups of $S_n$ with exactly one fixed point for each element all have the same fixed point.

Let $G$ be a subgroup of $S_n$ (where $n$ is a positive integer) such that each non identity element $g\in G$ has exactly one fixed point. Prove there is an element of $[n]$ that is fixed by every ...
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1answer
50 views

True or false simple algebra questions (centralizers, conjugacy classes, normal groups, abelian groups) [closed]

Can someone please verify my answers to the following questions? Note: This is NOT homework! Answer true or false to the following questions: Two elements of a group in the same conjugacy ...
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2answers
28 views

Permutations with Condition

I have looked at this old problem in my textbook: How many permutations $\pi \in S_n (n \geq 3)$ meet the requirement: $\pi (1) < \pi (2) $ or $\pi (1) < \pi (3)$? I am not sure how to ...
2
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2answers
40 views

Equations in groups

I want to solve an equation $$f(\sigma , \tau , \delta)=1$$ where $\sigma,\tau,\delta$ are elements from a given group $G$, and $1 \in G$ is the unit element. When I say solve I mean give sufficient ...
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$ G $ is satisfies the maximal permutizer condition if $ G $ is supersoluble?

Permutizer of a subgroup $ H $ of $ G $ is defined to be the subgroup generated by all cyclic subgroups of $ G $ that permute with $ H $, i.e. $ \langle x\in G \vert \langle x \rangle H = H \langle ...
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3answers
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Stanford math qual: Abelian groups $G$ satisfying $0\to \Bbb{Z} \oplus \Bbb{Z}/3\Bbb{Z} \to G \to \Bbb{Z} \oplus \Bbb{Z}/3\Bbb{Z} \to 0$

I am studying for my qualifying exams and came across the following question: Find all abelian groups $G$ that fit into an exact sequence $0\to \Bbb{Z} \oplus \Bbb{Z}/3\Bbb{Z} \to G \to ...
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1answer
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Minimal order of a group with a particular property

I fix an integer $n$. I am looking for a group $G$ for which there exist elements $g_1, \dots, g_n \in G$ and $h_1, \dots, h_n \in G$ such that $$ h_kg_k^{-1} \neq h_j g_i^{-1}$$ as long as ...
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1answer
54 views

Can two non-abelian groups have an abelian product or coproduct?

It is easy to see that two or more abelian groups must have an abelian product (and coproduct, since these constructions coincide in $\sf Ab$.) I'm not sure how to proceed with this; I just thought ...
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2answers
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$H\subseteq G$, $N\triangleleft G$, and showing $|[G:N]|$ is a prime number

Let $G$ be a finite group and let $N\triangleleft G$ a normal subgroup. It is given that if $H\subseteq G$ such that $N\subseteq H\subseteq G$, then $H=N$ or $H=G$. Show that $|[G:N]|$ is a prime ...
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1answer
75 views

Does anyone know the Burnside Matrices?

For $G$ a fine group with conjugacy classes $C_1,\dots C_k$ we introduced the Burnside Matrices $A_r$ where $1<r<k$ with entries: $$A_r := \Big(\sqrt{\frac{|C_t|}{|C_s|}}a_{rst}\Big)_{1\leq ...
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0answers
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Is there any “Brunnian-like” braid which is not a pure braid in $B_n$ with $n\geq 3$?

A braid $\beta$ is called Brunnian if it satisfies $\beta$ is a pure braid; $\beta$ becomes a trivial braid after removing any of its strands. Obviously in $B_2$, every braid satisfies condition ...
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0answers
37 views

For which integers $r$ is $\sigma ^r$ also a $k$-cycle? [duplicate]

Let $\sigma$ be a $k$-cycle in $S_n$. For which integers $r$, is $\sigma ^r$ also a $k$-cycle? I think I managed to prove that this is true iff $(k,r)=1$, but my proof was too long and not elegant ...
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67 views

Orbit of a Weight Vector?

Given some element $\phi$ of a representation $R$ of a group $G$, the orbit $G(\phi)$ of $\phi$ is defined as the set $g \phi \ \forall \ g \in G$. We can write every element of a given ...
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Is the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ isomorphic to $\frac{SO(3)}{H}$?

I have heard many times that the homotopy group of the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ and of the space $\frac{SO(3)}{H}$ are identical. I.e., $\frac{SO(3) \times Z_2}{H \times Z_2} ...
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1answer
92 views

Rank 3 permutation groups

Let $G \leq Sym(\Omega)$ be a finite permutation group of rank 3, $\alpha \in \Omega$ and $g,h \in G$ such that $x_1 := g(\alpha)$ and $x_2 := h(\alpha)$ are not equal. Now my question is: Is there ...
5
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1answer
66 views

Haar measure on SO(n)

I am interested in describing the group of special orthogonal matrices SO(n) by a set of parameters, in any dimension. I would also like to obtain an expression of the density of the Haar measure in ...
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1answer
21 views

Prove that if $a\in G^n$ and $g\in G$ then $g^{-1}ag\in G^n$

Let $G$ be a finite group and $n\in \mathbb{N}$. For all $a,b\in G$ there is: $(ab)^n=a^nb^n$. Define $G^n=\{g^n\ |\ g\in G\}$. Prove that $G^n$ is a subgroup of $G$ and that if $a\in G^n$ and ...
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1answer
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Let $G$ be finite group. if $A,B\le G$ with orders $4, 5$ respectively then $A \cap B$? [closed]

Let $G$ be finite group. If $A$ and $B$ are subgroups of $G$ with orders 4 and 5 respectively, what is $A \cap B$ ?
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1answer
42 views

Representations of the form $\varphi: G \rightarrow GL(V)$ vs $\phi: G \rightarrow Aut(A)$

Standard representation theory studies homomorphisms of the form $\varphi: G \rightarrow GL(V)$ where $V$ is a vector space. How much does the focus of representation theory change if one considers ...
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2answers
26 views

Group acting on $X$ and element of normal subgroup $H$ fixes an element of $X$ implies $H$ fixes all of $X$

A group $G$ acts on a set $X$ transitively and a normal subgroup $H$ fixes a point $x_{0} \in X$, i.e. $h \cdot x_{0}=x_{0}$ for all $h \in H$. Show that $h \cdot x = x$ for all $h \in H$ and $x \in ...
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2answers
66 views

Number of cycles in complete graph

How many number of cycles are there in a complete graph? Is there any relation to Symmetric group?
5
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2answers
76 views

$G=\langle x,y\ |\ x^{-1}y^2x=y^{-2}, y^{-1}x^2y=x^{-2} \rangle$ is torsion-free.

I have to prove that $G=\langle x,y\ |\ x^{-1}y^2x=y^{-2}, y^{-1}x^2y=x^{-2} \rangle$ is torsion-free. Some things about this group that I understand are first show that $M=\langle (xy)^2,x^2,y^2 ...
2
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4answers
83 views

Subgroup of $\mathbb{Z}$ generated by two positive integers

An exercise from Aluffi's Algebra book. Let $m,n$ be positive integers and consider the subgroup $\langle m,n\rangle$ of $\mathbb{Z}$ they generate. As a subgroup of $\mathbb{Z}$ it will be equal ...