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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Proof “correctness” : Cycle structure of conjugate permutations

My Algebra lecturer is a very strict about proofs(w.r.t Completeness , correctness and format ) more so than I have encountered in the past or any of my lecturers of the courses I am take concurrent. ...
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37 views

Symmetry groups of discrete functions

I'm looking for basic information about symmetry groups of discrete functions. It is difficult to search for such information, because searching for "symmetry group" gives results that refer almost ...
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44 views

Subsets of cyclic group with distinct pairwise differences

Given a number $m\in\mathbb N$, let $\mathbb Z_m=\{0,1,\dots,m-1\}$ denote the ring of integers modulo $m$ (although we won't need multiplication, so any cyclic group of order $m$ will do). Given a ...
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~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~

I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ...
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215 views

Conjugation Quandles and… “Quandle-Groups”? From quandles to Groups.

A quandle $(Q,*,/ )$ is a idempotent right-distributive and right invertible structure. 1) $a*a=a$ 2) $(a*b)*c=(a*c)*(b*c)$ 3) $(a*b) /b=(a/b)*b=a$ If we have a group $(G, \cdot, ...
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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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100 views

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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109 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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129 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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179 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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112 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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155 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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168 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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70 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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261 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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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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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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58 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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328 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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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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117 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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74 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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61 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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185 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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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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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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482 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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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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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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128 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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208 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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150 views

Conjugacy classes and irreducible representations of GL_2(q)

Let $q$ be a prime. (a) Find representatives for the conjugacy classes of $GL_2(q)$. (b) Show that $GL_2(q)$ has an irreducible complex representation of dimension $q$. (c) Compute the character ...
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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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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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87 views

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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214 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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572 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?