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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On solvable group and normalizer

" Let $G$ be finite group, $H$ is a subgroup of $G$, and $P$ is a Sylow-p Subgroup of $G$. If $N_G(P) \leq H$, show that $N_G(H)=H.$ " This problem appears in Martin Isaacs book under the chapter ...
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Showing that $G$ is abelian [duplicate]

Suppose $G$ is a finite group whose order is not divisible by 3. Suppose that $a^3b^3=(ab)^3$ for all $a,b$ in $G$. Then I wish to show that $G$ is abelian. $a^3b^3=(ab)^3=ababab \implies ...
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If $A \lhd P$ and $A = C_P(A)$, then $|P:A|$ divides $(|A| - 1)!$

This is problem 1.D.10 in Isaacs, Finite Group Theory. Let $A$ be maximal among the abelian normal subgroups of a $p$-group $P$. Show that $A = C_P(A)$, and deduce that $|P:A|$ divides $(|A|-1)!$ ...
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Why is $\pi_1(F_g)^{ab} = \Bbb Z \langle a_1, b_1, \ldots , a_g, b_g \rangle$?

I am told that for a surface with genus $g$, call it $F_g$, the abelianization of $\pi_1(F_g) = \langle a_1, b_1, \ldots , a_g, b_g \mid [a_1, b_1] \cdots [a_g,b_g] = e \rangle$ is $\pi_1(F_g)^{ab} = ...
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50 views

A Proof Check for A Theorem About Cyclic Groups, A.A. Albert

I was working with A.A. Albert's Fundamental Concepts of Higher Algebra and noticed I was able to complete one of the exercises without actually using all the assumptions. I imagine there must be ...
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2answers
97 views

Calculating the normalizer of a sylow p-subgroup

It seems that its explained pretty well online how to find a p-sylow subgroup, but Im having a hard time finding an explanation of how to find a p-sylow subgroups normalizer. Take a specific 2-sylow ...
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50 views

$\mathbb{R}^1$-bundle $\xi$ possesses Euclidean metric iff $\xi$ represents an element of order $\le2$

The set of isomorphism classes of $1$-dimensional vector bundles over $B$ forms an abelian group with respect to the tensor product operation. How do I see that a given $\mathbb{R}^1$-bundle $\xi$ ...
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150 views

I don't understand a Michael Aschbacher's proof

I can't understand so much the last paragraph (page 165) of the proof of the lemma 32.3 (pages 163-165) in M. Aschbacher, Finite Group Theory about Thompson Factorization. Here I post the statement of ...
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What non-abelian groups have a minimal permutation representation that acts transitively $\{1,2,\ldots, k\}$?

This question asks if a minimal permutation representation $\overline{G}$ of a group $G$ (that is, a subgroup $\overline{G} \le S_k$ is isomorphic to $G$ and $k$ is minimal with respect to this ...
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What does $\langle X\rangle$ mean?

My notes has the following. Let $G$ be a group. Let $J = \{G_i\}_{i∈I}$ to be the collection of all subgroups of $G$ containing the subset $X ⊆ G$. We note that $G$ is nonempty, since $G$ itself ...
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47 views

Understanding Abel-Ruffini

I'm wondering of anyone can point me towards a proof of why we can't have a quintic formula, using concepts from basic group theory. In particular, I understand that there is some connection with ...
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1answer
150 views

Is $n=8{,}574{,}796{,}230$ the smallest squarefree number $n$ with $gnu(n)>10^6$?

The number of groups of order $n$ (gnu(n)) can be calculated by a closed formula, if $n$ is squarefree. I could calculate the values with GAP, additionally, I programmed a version in PARI/GP, working ...
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39 views

Help justifying unique 2n-1 sum pairs as (2n-1, {3, 5,…,2n-3,2n-1}) [closed]

Need help for a long proof I am doing. I need to justify that if I examine the set of {Odd+Odd}, the unique pairs exist if I utilize {2n-1, {3,5...,2n-3,2n-1}) If we have: 3 5 7 9 ...
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3answers
61 views

Is this induced action transitive?

Let $G$ be a group of order $n$ and let $k$ be a smallest integer such that we have a injection from $G$ to $S_k$. Denote $\bar G$ as a image of $G$. Is the action of $\bar G$ on $\{1,2,...,k\}$ ...
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1answer
76 views

Prove that if a group $G$ has $|G| = 6$ then $G$ is isomorphic to either $\Bbb Z/6$ or $S_3$

Prove that if a group $G$ has $|G|$ = 6 then G is isomorphic to either $\mathbb{Z}_6$ or $S_3$. I have the proof by contradiction but I was wondering if there was a direct proof instead in case I ...
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1answer
41 views

Is there a non-trivial cyclic quotient group of a non-cyclic group? [closed]

Let G be a non-cyclic group and H be a non-trivial normal subgroup. Can we state that G/H is non-cyclic?
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Solution about a Michael Aschbacher exercise [edit]

I really need the solution of the exercise 11.4 (page 174) in M. Aschbacher, Finite group theory. Here is the text of the exercise. Let $G$ be a finite group, $p$ a prime, $\Omega$ a ...
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3answers
41 views

Order of a symmetry group of a quadrilateral

I am studying group theory. When reading text on the theory part, I do not feel difficult to understand the concepts, like what is a group, or what is Cauchy's Theorem. However, I often find it ...
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55 views

Show that a group of order $33$ cannot only have elements of order $11$ and $1$.

Attempt: Assume $G$ only has elements of order 11 and that there exists $x \in G$ such that $o(x) = 11$. Then $x^{o(G)} = e$. $x^{33} = (x^{11})^3 = e \implies x^{11}$ has order $3$. Contradiction.
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Finding an example of a set $G$ which is not a group

Suppose $G$ is a set and $\cdot$ is a binary operation on $G$ such that there exists an $e\in G$ such that $a\cdot e=a$ for a in $G$ and given $a\in G$, there is a $y(a)\in G$ such that $y(a)\cdot ...
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If $H$ is subnormal in $G$ and its index is a $\pi$-number, then $O^{\pi}(G) \le H$?

Let $\pi$ be a set of primes, then we call $n \in \mathbb N$ a $\pi$-number if it only contains prime divisor from $\pi$. If $G$ is a finite group, a $\pi$-group is a group whose order is a ...
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A finite group containing only elements of order 1 or 2 must have even order

Suppose $G$ is a finite group in which every element has order $2$ or $1$, and that $G$ contains some element different from the identity element. How can I explain that the order of $G$ must be even? ...
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1answer
27 views

Weyl group of $V_{4}$ in $S_{4}?$

Can anyone please explain what a Weyl group is and what a Weyl group of $V_{4}$ in $S_{4}$ is? I do not understand this exercise.
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Is my proof of $C_G(H) \le N_G(H)$ correct?

Let $x\in C_G(H)$. This means $xh = hx$ for all $h \in H$. Then $xH = Hx$ (This is that part I'm not so sure about). Hence, $x \in N_G(H)$, so that we have $C_G(H) \le N_G(H)$.
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110 views

Describe all groups of 8 elements

I try to find all the groups of 8 elements. I have found: $\mathbb{Z}_8$ $\mathbb{Z}_2 \times \mathbb{Z}_4$ $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$ quaternions I don't understand ...
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1answer
59 views

Showing that $H$ is normal [duplicate]

This question has been bugging me for a while. Suppose that $H$ is a subgroup of $G$ such that $Ha\not=Hb$ implies that $aH\not =bH$. I need to show that $gHg^{-1}\subset H$.(I don't mean a proper ...
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Irreducible invariant tensor spaces of the general linear group $GL_n(\mathbb{C})$

I am learning the representation theory of the general linear group $GL_n(\mathbb{C})$. As far as I understand, the way to decompose the $\nu$-fold tensor product space $ V^{\otimes \nu}=V \otimes V ...
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1answer
32 views

Conjugacy classes of a group [closed]

What is number of conjugacy classes in the permutation group $S_{6}$ I only knows that $o(S_6)=6!$
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1answer
102 views

What does $\widehat{\mathbb{Z}}(1)$ mean?

I just received a hand-written letter from a very famous mathematician, who in his email used the notation $\widehat{\mathbb{Z}}(1)$. For the longest time I thought that he meant ...
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Group of rotations of rational multiples of $\pi$

Consider the euclidean plane $\mathbb{R}^2$ and denote the rotation around the origin with angle $\theta$ by $$R_\theta:\mathbb{R}^2\rightarrow \mathbb{R}^2$$ Now consider a fixed angle $\theta_0\in ...
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What can we say about $D_{2n}$ and $D_n$ if $n$ is even?

We know that if $n$ is an odd natural number then the dihedral group $D_{2n}$ is isomorphic to the group $D_n\times \mathbb{Z}_2$ where $D_m$ is the dihedral group of order $2m$, which has ...
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1answer
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In $S_5$, we have $aba^{-1}=b^2$, $b=(12345)$, find $a$.

In $S_5$, we have $aba^{-1}=b^2, b=(12345)$, find $a$. I have tried different ways to substitute/rearrange, but none of them worked.
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2answers
87 views

Proving a group of order $77$ has a subgroup of order $7$ without Sylow theorem.

The question is Show if $G$ has order 77 then $G$ has a subgroup of order 7. Without using Sylow Theorems. Attempt sketch: Let $x \in G$. By Lagrange's theorem the order of $x$ is either $1, ...
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Finding $N(D_{4})/D_{4}$ for $D_{4}$ in $D_{16}$

I want to find $N(D_{4})/D_{4}$ where $N(D_{4})$ is the normalizer of $D_{4}$ in $D_{16}$. I'm not too clear on what the normalizer of $D_{4}$ in $D_{16}$ Is there a nice way to find ...
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2answers
160 views

Rubik's Revenge Cube in GAP

I'm trying to create the Rubik's Revenge (4x4x4 cube) group in GAP . Take the following net of the 4x4x4 cube with each sticker labelled with a number. The front, left, upper, right, down, and back ...
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1answer
40 views

Show that $\varphi:KN\rightarrow K$ is a homomorphism

I am having trouble with a particular mapping, determining whether or not it is a homomorphism. I know that $K,N$ are normal subgroups of $G$, $K\cap N=\{e\},$ and $KN=NK$ where $$KN=\{kn|k\in K, ...
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1answer
38 views

Prove $KN\cong K\times N$ if $K\cap N=\{e\}$

Let $K$, $N$ be normal subgroups in $G$. I would like to show that if $K\cap N=\{e\}$, then $KN\cong K\times N$. My original thought was to invoke the first Isomorphism theorem. Therefore i would ...
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Key Result in Group Theory

I've been going about doing one of the key results in Group Theory: Given a group $G$, for all subgroups $H \leq G,$ $o(H)$ is a multiple of $o(G).$ I have a feeling a good start to this would be ...
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Units in Semiperfect Skew Group Rings

Let $k$ be a field and $S$ the ring $k[[x_1,\ldots, x_n]]$. Let $G$ be a finite subgroup of $GL_n(k)$ that does not contain any nontrvial pseudo-reflections and such that $|G|$ is invertible in $k$. ...
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No simple group of order 2016

The year 2016 is coming up, so it makes me wonder: how do you prove that there is no simple group of order 2016? (without invoking the 10,000 page theorem on the classification of simple groups)
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The projective special linear group $PSL(2,\mathbb F)$ acts equivalent on $\mathbb F^2$ and $\mathbb F_{\infty} = \mathbb F \cup \{\infty\}$

This question is closely related to this one. Again consider the group $G := PSL(2, \mathbb F)$ over some field $\mathbb F$. Then as written in the other post, there are two natural actions ...
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Residually Finite Braid Group

In Braid Groups of Kassel, Turaev, it mentions that $\mathcal{B}_n$ is a residually finite group. The definition that they give as a residually finite group is a group $G$ such that for each $g\in ...
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1answer
51 views

Frattini subgroup of a finite elementary abelian $p$-group is trivial

I would like to improve my proof of the following result: If $H$ is a finite, elementary abelian $p$-group, then $\Phi(H) = 1$. Here, $\Phi(H)$ is the Frattini subgroup, defined as the ...
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If $G / N$ is a $\pi$-group, then $O^{\pi}(N) = O^{\pi}(G)$?

Let $G$ be a finite group and $\pi$ a set of primes, then $O^{\pi}(G)$ denotes the smallest normal subgroup such that $G / O^{\pi}(G)$ is a $\pi$-group (i.e. a group such that its order is only ...
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1answer
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Find the left cosets of subroups of $S_3$

So I am struggling to understand the definition of a coset. If I have the following symmetric group $S_3=\{1, \sigma, \sigma\tau, \sigma\tau^2, \tau, \tau^2\}$, where ...
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Is $gnu(n)<n$ always true for cubefree $n>1$?

Let $gnu(n)$ be the number of groups of order $n$. If $n$ is cubefree, (there is no prime $p$ with $p^3|n$), does the inequality $gnu(n)<n$ always hold for $n>1$ ? According to GAP, upto ...
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35 views

Colored beads on a loop

Suppose we have $p$ beads of $n$ different colors on a loop. $p$ is a prime number and we consider the loop to be the same if one is a rotation of the other. Then how many distinct beads are there? By ...
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1answer
75 views

How to write dihedral group in cycle notation?

Since each symmetry can be thought of as a permutation of the vertices, the elements of $D_n$ can be thought of as elements of $S_n$. So I'm wondering if there's a systematic way that we can always ...
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2answers
39 views

Monomorphism between abelianizated groups

I want to find an example of a group monomorphism, $$\begin{matrix}\phi:&G_1&\longrightarrow&G_2 \end{matrix}$$ such that, ...
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2answers
111 views

Order of a generator element of Free Group in GAP

I am using the FreeGroup function in GAP ...