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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Can $\mathrm{PGL}_2$ be viewed as an affine algebraic group?

I was just wondering whether or not it is possible to view $\mathrm{PGL}_2$ as an affine algebraic group.
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34 views

Where do I use that $G$ is a permuation group?

This is about question $4.1.7$ from Dummit and Foote, and also related to my previous question. The question is (summarised a bit): Let $G$ be a transitive permutation group on a finite set $A$. ...
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80 views

Defining a coproduct in $\mathsf{Grp}$ using group presentations

I've encountered this exercise in Aluffi's Algebra: Chapter 0. It might be helpful to say that the book doesn't introduce functors at this stage,, and that the book defines a presentation of a group ...
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1answer
90 views

Proving a certain lemma about subgroups of $A_n$

In proving $A_n$ is simple for $n\neq4$, my teacher established the cases 1, 2, 3 as obvious, then proved the case 5, and proceded by induction on the rest. In the midst of that induction, he stated ...
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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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56 views

Matrix and Abelian groups question

Let $A$ be a Matrix: $$ A=\begin{pmatrix} 1 & 2\\ 4 & 1 \end{pmatrix} $$ Let $f\colon v\to Av$ be a homomorphism from $Z^2$ to $Z^2$. Find a base $(v_1,v_2)$ to $Z^2$ and $2$ integers ...
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37 views

Let G be a group, $N<K<G$ and $N\trianglelefteq G$. Prove that $K/N \trianglelefteq G/N$

Let G be a group, $N<K<G$ and $N\trianglelefteq G$. Prove that $K/N \trianglelefteq G/N$ What I have tried is: Note that $1\in G$. So $1\in K$ and $N1=N\in K/N$ which shows that $K/N$ is ...
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Computing Factor Group

I am reading John Fraleigh's First Course in Abstract Algebra, $\S$36 on the Second Isomorphism Theorem which says that if $H < G$ and $N \triangleleft G$, then $$(HN)/N \cong H/(H \cap N).$$ He ...
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30 views

Group theory exercise from “Rational Points on Elliptic Curves”

Let $A$ be an abelian group and for every $m \geq 1$, let $A_m$ be the set of elements $P$ of $A$ such that $mP=0$. Now, suppose that $A$ has order $M^2$ and that for every integer $m$ dividing ...
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84 views

Quantity of elements of order $d$ in $Z_n$, with $d \mid n $

I'm studying for an exam and I can't answer this problem. I'd appreciate a hint. What I've got so far: Let $x$ be an element or order $d$. Then, $x\cdot d \equiv 0 \pmod n \Rightarrow n \mid x\cdot ...
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40 views

Determine the isomorphism class of $\mathbb Z^3 / M$ for the subgroup $M$ of $\mathbb Z^3$generated by $(13,9,2),(29,21,5),(2,2,2)$

The problem seems not so hard. My confusion rise from the statement in the solution above that "This question is equivalent to reducing the matrix via row and column operations". Please see the ...
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1answer
79 views

Finitely generated abelian group with certain properties

Problem Characterize all finitely generated abelian $G$ such that every proper subgroup of $G$ is cyclic, $G$ contains exactly two proper subgroups, and for each pair of subgroups $S$,$T$ in $G$ ...
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1answer
65 views

Examples of irreducible representations

Which of the following representations are irreducible? 1) The tautological representation of $D_n$ on $\mathbb{R}^2$ 2) The action of $U(1)$ on $\mathbb{C}$ by multiplication 3) The ...
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28 views

Unit group of an algebraically closed field is divisible

In the lecture notes on Valuation theory, in Ex. $1.16$ on page $11$ we are asked to show that: If $k$ is an algebraically closed field, then $k^{\times}$ is a divisible abelian group. Isn't $k ...
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Conjugates of Sylow $p$-groups in $GL_3(F_p)$

In this list of review questions, there is the following question about $GL_3(F_p)$. Question 1.38. Let $G$ be the group of invertible 3 × 3 matrices over $F_p$, for $p$ prime. What does basic ...
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40 views

Derived subgroup of $S_n$ and $D_n$

I know that Derived/Commutator subgroup of $S_3$ is $A_3$ and commutator subgroup of $D_4$ is cyclic of order $2$. But What about derived groups of $S_n$ and $D_n$? How can I calculate them?
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What does $(G:G')$ mean?

I'm trying to teach myself some ring theory from a book, and have come across this sentence: "There are $(G:G') = 4$ linear characters" where $G$ is a group, and $G'$ is the derived group. I ...
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41 views

action of a subgroup on a metric space

If $G$ acts properly and cocompactly by isometries on the metric space $X$ and if $H$ is a subgroup of $G$. Does $H$ act properly and cocompactly by isometries on a subspace of $X$?
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31 views

transitive subgroups of the symmetric group on $2(d+1)$ elements: can I always do at least $d$ permutations?

I have a set of $2(d+1)$ elements which are labelled as pairs $\{e_i, a_i\}_{i=1}^{d+1}$, transforming under some transitive subgroup of the symmetric group. This can be thought of as a regular ...
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46 views

Origin of the term “derived subgroup”

The commutator subgroup G', generated by all commutators of a group G, is also called the derived subgroup. Why is this; are there any concrete analogies with analytical derivatives (apart from the ...
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56 views

Left and right action?!

The adjoint of the adjoint representation $Ad^* : G \times \mathfrak{g}^* \rightarrow \mathfrak{g}^*, (g,x) \mapsto Ad^*_{g}(x)$ is a group action on the dual space of the Lie algebra. Now, we said ...
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2-Frobenius groups of order $2^{10}.3^5.5.11$

A group $G$ is called a 2-Frobenius group if $G=ABC$, where $A$ and $AB$ are normal subgroups of $G$, $AB$ is a Frobenius group with kernel $A$ and complement $B$ and $BC$ is a Frobenius group with ...
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Different descriptions of the Baumslag-Solitar groups using affine groups

On page 101 of this paper of Laurent Bartholdi (which is an online documentation of the FR package for GAP which allows GAP to manipulate groups generated by automata) he gives a different description ...
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Prove that all normal subgroup definitions are equivalent.

given $ N<G $ I need to prove that all of the below are equivalent: 1) for each $g \in G$ , $n \in N$ $gng^{-1} \in N $ 2) for each $g \in G$ $gNg^{-1} = N $ 3) for each $g \in G$ $gN = Ng$ ...
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1answer
45 views

What is the probability of 2 random matrices generate a free group?

Let A,B $\in GL(2,Z)$, then what is the probability of $<A,B>\cong F_2$? By probability, I mean the haar measure on $GL(2,Z)^2$. I already know what if we replace $\mathbb{Z}$ with ...
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Presentation of the additive group of the rational numbers

We know that $\mathbb{Q}\cong\mathbb{Z}\times\mathbb{Z}/\sim$, where the isomorphism is a ring isomorphism and the equivalence relation is defined as $$(a,b)\sim(c,d)\Longleftrightarrow ad=bc$$ Then ...
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The galois group of a polynomial of degree 3 is either $A_3$ or $S_3$

Hungerford -Algebra p.271 Let $E/F$ be a Galois extension where $E$ is a splitting field for a separable irreducuble polynomial $f$ over $F$ whose roots are $a_1,a_2,a_3$. Let ...
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Any mathematical software to verify no unique products.

Let $D^3= D \times D \times D$ where $D = D_\infty$ where we see $D$ as the group generated by $\mathbb{Z}$ and element $0^*$ of order $2$ such that $0^*n0^*=-n$ for all $n \in \mathbb{Z}$. Letting ...
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$H$ and $K$ are subgroups of $G$.Recall that $HK=\{hk:H\in H,k\in K\}$ Show that $|HK|=|H||K|/|H\cap K|$

Let $G$ be a finite group, and let $H$ and $K$ be subgroups of $G$.Recall that $HK=\{hk:H\in H,k\in K\}$ Show that $|HK|=|H||K|/|H\cap K|$
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1answer
106 views

greatest common divisor of a cyclic group generator [duplicate]

I have been struggling with this for a very long time-in fact for a few days and it's preventing me from progressing-a severe hurdle. Mainly, 1) what is $$\left \langle a^{k} \right \rangle$$? Is ...
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1answer
20 views

Equivalence relation for equidistant points in the plane

Let an equivalence relation $R$ on the plane be: $(a,b)\in R$ if $a$ and $b$ are equidistant from the origin. Well, $(a,a)\in R$ and also, $(a,b)\in R$ means $a$ and $b$ have the same distance from ...
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1answer
51 views

Groups occuring as derived subgroups.

I want to solve this problem but I have no idea how to start it. If you know please hint me, thanks. Suppose that $G$ is a group that has subgroup which is cyclic, characteristic and not in the ...
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1answer
48 views

An $RO$-group which is not $O$-group

I was thinking of some example for an Right ordered group ( $RO$-group) which is not an $O-$group (Ordered group) i.e. not left ordered. I guess looking in matrix groups will be fruitful but how to ...
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Can every torsion-free nilpotent group be ordered?

I know that a torsion-free abelian group can be ordered and have done two proofs for that too. But the next two question that popped up in my mind were- Can every torsion-free nilpotent group be ...
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2answers
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$H,N(H)$ are subgroups of $G$ show that $H\lhd N(H)$

Let $G$ be a group and $H$ subgroup of $G$, $N(H):=\{g\in G; gHg^{-1}=H\}$ $N(H)$ is also subgroup of $G$. I need to prove that $H$ is a normal subrgoup in $N(H)$ Attempt: $H\lhd N(H) \iff ...
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2answers
56 views

Show that $N(H):=\{g\in G; gHg^{-1}=H\}$ is subgroup of $G$

Let $G$ be a group and $H$ subgroup of $G$, $N(H):=\{g\in G; gHg^{-1}=H\}$ I need to prove that $N(H)$ is subgroup of $G$. It's almost the same question like :$\forall g \in G, gHg^{-1} = H ...
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1answer
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What are all the $k$-dimensional unimodular subspaces of $\mathbb{Z}^n$? [duplicate]

I am trying to prove the following assertion: The set of subgroups of $(S^1)^n$ which are isomorphic by an element of $\operatorname{Aut}((S^1)^n)$ to the standard copy $(S^1)^k$ is naturally ...
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2answers
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Show that if $G$ is a finite cyclic group then $G^*$ is cyclic and $o(G)=o(G^*)$

Let $G$ be a group and $G^*$ be the group of all homomorphisms from a group $G$ to the set $\mathbb C^*$ i.e the group of all non-zero complex numbers. Show that if $G$ is a finite cyclic group then ...
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Question about Eigenvalues of group elements

The quaternion algebra is given by $\mathbb{H}$ = $\{a+bi+cj+dk \mid a, b, c, d \in \mathbb{R}, i^2 = j^2 = k^2 = -1, ij = k = -ji\} := \{z_1+z_2j \mid z_1, z_2 \in \mathbb{C}\}.$ I consider the ...
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Prove there generally is no isomorphism between $R[x]/(x^2-a)$ and $R^2$

I have a ring $\mathbf R =(R, +, -, ., 0, 1)$ (note that there is no invers for multiplication, $R$ is not $\mathbb R$, it is any set for the given algebra). How do you prove that the following does ...
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Question about working in modulo?

This question is in essence asking for understanding of a step in Fermats theorem done Group style. For any field the nonzero elements form a group under field multiplication. So let us take the ...
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1answer
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Matrices $P$ such that $A$ is symmetric $\Longrightarrow $ $PAP^{-1}$ is symmetric

Let $M_n(\mathbb{R})$ be the (vector) space of all $n\times n$ matrices over $\mathbb{R}$. Let $Sym_n(\mathbb{R})$ denote the subspace of symmetric $n\times n$ matrices. $GL(n,\mathbb{R})$ acts on ...
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Image of a normal subgroup under automorphism is the normal subgroup

Let $G$ be a finite group. Let $H$ be a normal subgroup of $G$ such that the order of H and the index of $H$ in $G$ are relatively prime. Let $f$ be an automorphism of G and let $J = f(H)$. Prove ...
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Books about braid theory

I'm looking for books that talk about braid theory, in the sense of braid groups mostly, and not too advanced, if possible. With material understandable for an undergraduate. Thanks for any ...
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If $G_{ab}$ is cyclic then $G$ is cyclic

In my notes I have the following theorem: Let $G$ be a (nilpotent?) group. Suppose that $G_{ab}$ is cyclic. Then $G$ is cyclic. Actually I don't know if the hypothesis that $G$ is nilpotent is ...
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Show that $\overline \varphi (a Z (D_4)) = Id$

Consider $$\begin{align}\overline \varphi : \frac{D_4}{Z(D_4)} &\to \frac{D_4}{Z(D_4)} \\aZ(D_4) &\mapsto xax^{-1}Z(D_4)\end{align}$$ where $$D_4 = \{id, \alpha, ...
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1answer
85 views

Group Acting on a Ring

What would be the definition for a group action on a ring? I could not find one online. Would this be acceptable? A group action of a group G on a ring R is a map from G x R to R defined by g(r)=g.r ...
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1answer
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How to determine $\left|\operatorname{Aut}(\mathbb{Z}_2\times\mathbb{Z}_4)\right|$

I know that the group $\mathbb{Z}_2\times\mathbb{Z}_4$ has: 1 element of order 1 (AKA the identity) 3 elements of order 2 4 elements of order 4 I'm considering the set of all automorphisms on this ...
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1answer
34 views

Why is ${\bf N}\otimes\bar{\bf N} \cong{\bf 1}\oplus\text{(the adjoint representation)}$?

I just watched this lecture and there Susskind says that $${\bf N}\otimes\bar{\bf N} ~\cong~{\bf 1}\oplus\text{(the adjoint representation)}$$ for the Lie group $G= SU(N)$. Unfortunately, he does ...
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1answer
58 views

Show that $\det \rho(g)=1$ for elements g of odd order and $\det \rho(g)=-1$ for elements g of even order.

$G$ is a finite group. $\rho$ is a representation of $G$, then $\rho \mapsto \rho(g)$ is a $1$-dimensional representation. Show that if $\det \rho(g)=-1$ for some $g \in G$ then $G$ has a ...