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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A sequence of subsets of an infinite group.

Is there an infinite group $G$ such there is not any sequence $(A_n)$ of its subsets such that always $$A_n=A_n^{-1}, \quad A_{n+1}A_{n+1}\subsetneqq A_n$$ ?
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Does there exist an infinite solvable group with no normal abelian subgroups?

This is impossible if $G$ is finite and solvable, because then $G$ has a (nontrivial) minimal normal subgroup $A$, which can be shown (using a trick) to be abelian. I'm trying to mimic the same ...
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Three-Dimensional Heisenberg group

Let $H$ be the three dimensional Heisenberg group of quantum mechanics Describe all the conjugacy classes in $H$ as subsets of $\Bbb{R}^3$. Attempt: I was able to find the inner automorphism of the ...
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90 views

Does this set can say more about group?

Let $G$ be a finite group, set $\Omega=\{H\leq G\mid N_G(H)=H\}$ It is easy to observe followings; $|\Omega|=1$ if and only if $G$ is nilpotent as normalizer grows in nilpotent groups. For $H\in ...
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96 views

Subgroups of Symmetric Group $S_4$ and Isomorphism

During my Algebra class we were given this exercise to solve at home, but I couldn't find any solution and I also did not really get the one our teacher gave us. So, the text was: Given G = S4 = ...
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125 views

Abelization of symmetric groups and its subgroups of bounded support

For $X$ a finite set one can show easily that the abelization of the symmetric group on $X$ is given by the group of order $2$ and that the commutator subgroup of $\operatorname{Sym}X$ is given by the ...
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113 views

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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38 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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219 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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101 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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78 views

Clarifying Semi-Direct Products: Example

I'm working through some questions on semi-direct products, and although I can work out these problems (for the most part), I usually have trouble completing them. I have identified some of the things ...
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117 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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131 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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184 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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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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156 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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169 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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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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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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204 views

How “bad” can presentation of the trivial group get?

These questions are sort of preliminary questions and reference requests for a project I am doing. Lets say, for concreteness, that $R$ is a set of words in the free group of rank two and that ...
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263 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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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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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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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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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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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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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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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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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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186 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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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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506 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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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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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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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 ...