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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Unicity inner automorphism symmetric group

I need to prove that the center of the symmetric group $S_n$ with $n\geq 3$ is trivial. I chose to use Lagrange's theorem: bearing in mind that the center is a subgroup, it is easily found that ...
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363 views

Counting simple quadrilaterals in a rectangular lattice.

I've been trying to make an algorithm to find the number of all possible simple quadrilaterals in a N*M lattice. I already have a brute force solution but since this is a Project Euler problem I ...
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Find all subgroups of $\mathbb{Z_2} \times \mathbb{Z_2} \times \mathbb{Z_4}$ isomorphic to the Klein $4$-group - Fraleigh p. 110 Exercise 11.12

I tried to fill in the steps but I'm still confounded by this solution. $|\mathbb{Z_2} \times \mathbb{Z_2} \times \mathbb{Z_4}| = 2\cdot2\cdot4$. $order(\mathbb{Z_2} \times \mathbb{Z_2} \times ...
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103 views

Order of the Normalizer of a Sylow $p$-subgroup in $S_{p}$

Question: Let $p$ be a prime and $P \leq S_{p}$ with $|P| = p$. Prove that $|N_{S_{p}}(P)| = p(p-1)$. I have already solved this problem, but I have come across a proof that is much more elegant than ...
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119 views

Application of Sylow Theorems.

I have a group theory exam coming up so I was reviewing through my textbook and I stumbled across a problem that looked interesting: Let $p < q$ be two primes and $G$ be a group of order $pq^3$. ...
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168 views

Show that $|G|=1000$ is not simple

I think I've done this, but I just want to check it's proved enough. So $|G|=1000=8\cdot125$. Then $n_5\in\{\text{factors of 8}\}$ so $n_5\in\{1,2,4,8\}$ congruent to $1\bmod5$. $2$, $4$, $8$ are ...
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111 views

Role of the discrete subgroups of Lie groups

This is a question I don't believe is too vague to admit a sensible answer: In what directions can the structure theory of the discrete subgroups of real and p-adic Lie groups applied? What ...
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47 views

Algorithm to write an element of $SL_2(\mathbb{Z})$ as a product of $S, T^n$

It is well-known that $SL_2(\mathbb{Z})$ is generated by $S = \left( \begin{array}{ccc}0 & -1 \\1 & 0 \end{array} \right), T = \left( \begin{array}{ccc}1 & 1 \\0 & 1 \end{array} ...
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152 views

Smallest normal subgroup making quotient abelian, nilpotent, solvable

Given a finite group $G$ that is not abelian, nilpotent, or solvable, what is the smallest normal subgroup $H$ in each case such that $G/H$ is abelian, nilpotent, or solvable (respectively)? In ...
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163 views

Normality of a subgroup

How is normality of a subgroup $H$ of $G$ reflected in the table that displays the operation on cosets ? What I did: Let $G$ is a group of $8$ elements as $G= \mathbb{Z}_8$ And let $H$ be the ...
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68 views

Is this proof that $|S_n|=n!$ correct?

I've always heard that it's trivial that $|S_n|=n!$ where $S_n$ is the symmetric group of degree $n$. Now, my proof was the following: consider $I_n=\{1,\dots,n\}$ and consider ...
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89 views

Embedding $\mathbb{G}_a$ into $GL_2$

Let $k$ be an algebraically closed field of characteristic $p$. I'd like to find interesting examples of closed embeddings $\mathbb{G}_a(k)\hookrightarrow GL_2(k)$, where $\mathbb{G}_a(k)$ is ...
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251 views

Show $G$ is isomorphic to the multiplicative group $\mathbb{R}^\times$

Let $G=\{x\in\mathbb{R}\mid x>0 \text{ and }x\neq 1 \}$ and define $*$ on $G$ by $a*b=a^{\ln b}$. Show $G$ is isomorphic to the multiplicative group $\mathbb{R}^{\times}$. I need to ...
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167 views

How many compatible group structures does a topological space admit?

Suppose I have a topological space $(G,\tau)$ and am interested in whether there exists a topological group $(G,*,\tau)$. In other words, can we assign a binary operation $*$ to $(G,\tau)$ which ...
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77 views

Haar measure on transversals

It is well known that there exists an invariant Haar measure on Locally compact group $G$. Haar measure on coset space and double coset space with respect to a closed subgroup $H$ and a compact ...
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123 views

Finitely Presented Group

Let the descending central series of a group $G$ be $$G=\Gamma^1\geq\Gamma^2\geq\cdots$$ where $\Gamma^{q+1}=[G,\Gamma^q]$ for which $\Gamma^q/\Gamma^{q+1}$ is an Abelian group. Suppose that $G$ is ...
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49 views

Generating A_n using involutions — minimum number of multiplications that are sufficient?

What is the smallest $k = k(n)$ for which the following holds? There is a set of involutions $I \subset A_n$ such that for every $a \in A_n$, there exist $i_1, \ldots, i_k \in I$ such that $i_1 i_2 ...
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55 views

Is $G_\Delta$ acting transitively on $\Delta$? ($\Delta$ is an imprimitive block )

Suppose $G$ acts imprimitively (but transitively) on set $X$ and let $\Delta$ be some block of imprimitivity. It should be clear, that $G_\Delta$ is acting transitively on $\Delta$, but I cannot see ...
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303 views

torsion subgroup of a finitely generated nilpotent group is finite.

Prove that the torsion subgroup of a finitely generated nilpotent group is finite. More generally, in any group with "almost" no torsion all periodic subgroups are finite. Here "almost" means that ...
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What is the smallest $n$ for which the usual “counting sizes of conjugacy classes” proof of simplicity fails for $A_n$?

A common proof of the simplicity of $A_5$ proceeds as follows. First, note that a normal subgroup is always a union of conjugacy classes. Also, a subgroup has order dividing the size of the group by ...
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Subgroup consisting of unipotent elements centralizes a flag

Is there a reasonably elementary and short proof that a subgroup of consisting of unipotent matrices over a field centralizes a flag? By elementary, I mean accessible to students who have had a ...
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76 views

Abelian group whose quotient is $p$-divisible

Let $G$ be an abelian group of finite torsion-free rank which has no $p$-divisible subgroup. The group $G$ has a free subgroup $F$ of finite rank and the quotient $G/F$ is $p$-divisible. Can be proved ...
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115 views

Icosahedral symmetry

Is there a clear and correct reference for full icosahedral symmetry that includes a presentation and its action on vertices, edges, and faces? The Wikipedia article for icoshedral symmetry gives ...
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73 views

Showing $\mathbb{Z}_{2} * \mathbb{Z}_{3} \cong\ (a, b\ |\ a^2 = b^3 = e)$

Let $G = (a, b\ |\ a^2 = b^3 = e)$. I recognize there must be an epimorphism $\phi : G \rightarrow \mathbb{Z}_{2} * \mathbb{Z}_{3}$ (the free product) by the Van Dyck theorem, but I must show an ...
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60 views

What is the structure of the Coxeter groups of type $\text{D}_n$

I am curious on the structure of the Coxeter group $G$ of type $\text{D}_n$. Here I let $\{e_1,\cdots,e_n\}$ be the standard basis of the vector space $\mathbb{R}^n$. Then I choose ...
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183 views

Generation of left ideals in group rings

This looks like an elementary exercise on group rings (I heard it somewhere), nonetheless it seems to be non-trivial to me. Any references much appreciated. Suppose that we are given an (infinite) ...
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208 views

What is the intuition of conjugacy classes?

How can I fully understand what are conjugacy classes are in groups? I know the definition, that $a$ and $b$ are conjugate if $gag^{-1}=b$ for some $g\in G$. But what is the intuition? Using a ...
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95 views

Find all the central extensions of $\Bbb{Z}_2$ by $\Bbb{Z}_2 \times \Bbb{Z}_2$.

Find all the central extensions of $\Bbb{Z}_2$ by $\Bbb{Z}_2 \times \Bbb{Z}_2$. At first I was going to say that, since central extensions mean a trivial homomorphism, there must be only one: ...
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83 views

Proof that if $H \triangleleft G$ and $G/H$ is abelian, then $G' \le H$

I'd like some input on one part of my attempted proof for the following result. The other part I feel good about. "If $H \le G$ is any subgroup, show that $G' \le H$ if and only if $H \triangleleft ...
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75 views

What about non finitely generated groups?

Finite groups and finitely generated groups are intensively studied, but are there interesting investigations on non finitely generated groups? I already know some references for abelian groups, so I ...
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462 views

Finding the dimension of the symplectic group

How do you find the dimension of the symplectic group $Sp(2n,\mathbb{R})$? $Sp(2n,\mathbb{R})\subset Gl(2n,\mathbb{R})$ is the group of invertible matrices $A$ such that $\omega = A^T\omega A$, where ...
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Irreducible representation of $S_4$

Could one please point out an irreducible representation of degree 2 of the group $S_4$. Thank you.
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96 views

Prime divisor in the Automorphism group

Suppose that $G$ is a non-abelian simple group and $r$ is a prime number such that $r$ not divide $|G|$. It is a question for me that whether $r$ can divide |Aut($G$)|?
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What is the relationship between solvable, nilpotent and transfer homomorphism?

I know some facts such as: Nilpotent groups are solvable, $p$-groups are nilpotent, a finite group whose order is a product of distinct primes is solvable, and finite groups are nilpotent if and only ...
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Problem 4.3, I. Martin Isaacs' Character Theory

Let $G=H\times K$ be the direct product of finite groups. Let $\varphi\in Irr(H)$ and $\eta\in Irr(K)$ be faithful. Show that $\varphi\times\eta$ is faithful if and only if $(|Z(H)|,|Z(K)|)=1$. Here, ...
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322 views

Semi-direct product isomorphic to direct product

I would like some help on the following problem from anyone who would like to help. Let $f: H \to G$ be a group homomorphism. For $h \in H$, define $\rho(h) = \phi_{f(h)} \in Aut(G)$. The situation ...
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114 views

Has every finite group a minimal presentation ?

For a finite group $G$ let $d(G)$ be the minimal number of generators of $G$ and let $r(G)$ be the minimal number $r$ such that $G$ has a finite presentation with $r$ relators. Call a presentation ...
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is a group that eliminates quantifiers in the group language Abelian?

Let $G$ be a group. Consider it as a structure in the language of groups $L=(e,\cdot, (-)^{-1})$. Suppose that $G$ eliminates quantifiers in this language. Is it true that $G$ must be Abelian then? ...
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200 views

Easy proof of trivial fusion implies normal p-complement

Theorem: Suppose G is a finite group with Sylow p-subgroup P. Then the following are equivalent: The set K of elements of G of order relatively prime to p (the p′-elements) form a subgroup If A and ...
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138 views

finitely presented groups of exponential growth

Does there exists a finitely presented group of exponential growth which does not contain free sub-semigroups (of rank $\geq 2$)?
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259 views

Matrix group as subgroup of $GL(n,\mathbb{R})$ or $GL(n,\mathbb{C}$)?

As far as I understand, one has (at least?) two choices to introduce infinite matrix groups: Either, one can say they are all subgroups of the general linear group over the complex numbers numbers ...
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Ways to think about one-relator groups

What are some intuitive ways to think about one-relator groups? I am aware of the Freiheitsatz, and Bass-Serre theory. What I'm interested in are ways people who work extensively with one-relator ...
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213 views

Clarification: intersection of a finite number of subgroups of finite index and Poincaré

From Scott's book Group Theory $1.7.10.$ (Poincaré) The intersection of a finite number of subgroups of finite index is of finite index. My question is: Did Poincaré prove the Theorem as stated ...
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Is there a formula for $|\langle H\cup K\rangle|$, if $H$ and $K$ are nonempty subsets of a finite group $G$?

I know that if $H$ and $K$ are subgroups of a finite group $G$, then $|HK|=\frac{|H||K|}{|H\cap K|}$. Is there a formula for $|\langle H\cup K\rangle|$, if $H$ and $K$ are nonempty subsets of a ...
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239 views

Exercise 6.3.16 from Scott, Group Theory

How to demonstrate that a simple non-abelian group of odd order has order divisible by the cube of its smallest prime divisor?
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Looking for: a subgroup of an uncountable simple group of countable index

Consider the simple group $A_\lambda$, the alternating group on the set $\lambda$, which I will assume has regular cardinality. Recall that this is the smallest subgroup of all permutations of ...
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170 views

Is $H^2$ weakly closed in $H$? Isaacs Finite Group Theory, exercise 8B.6

I'm trying to solve exercise 8B.6 on page 249 of Isaacs's Finite Group Theory textbook (the second question in a series; this is the third as question here). I have an idea, but it doesn't quite ...
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When does PSL(5,q) have order exactly divisible by a specific odd power of 5?

In a misguided attempt at answering a question on divisibility of simple group orders, I looked at $\newcommand{\PSL}{\operatorname{PSL}}v_p(|\PSL(p,q)|)$ which went smooth enough for $p=2$ and $p=3$, ...
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558 views

What does a Cayley table tell?

Other than showing if a group is abelian and the order of the group, what else does a Cayley table indicate?
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146 views

a split exact sequence of finite groups

Suppose G has a cyclic normal subgroup $\langle a\rangle$ of order $m$ and prime power index $s$ such that $m$ and $s$ are relatively prime. Then the following exact sequence splits: $$1 ...