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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Number of necklaces of 16 beads with 8 red beads, 4 green beads and 4 yellow beads

The question is as stated in the title up to symmetries of $D_{16}$. I know this has to do with the following two formulas: If $G=D_{16}$ is the group acting on the set $S$ of different necklaces, ...
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73 views

Cyclically reduced words in free groups

Let $F$ be a free group on $\{x_1,x_2,\cdots\}$, $w$ a word in $F$. Then $w$ is a finite expression in $x_i$'s and their inverses. By cancellation, $w$ can be reduced to simplest expression. Let $w=...
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99 views

Corrolary 15 in *Abstract Algebra* by Dummit and Foote

On page 94, the Corollary states: If $H$ and $K$ are subgroups of $G$ and $H \leq N_G(K)$, then $HK$ is a subgroup of $G$. In particular, if $K \trianglelefteq G$ then $HK \leq G$ for any $H \...
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98 views

Geometric meaning of normal in group theory?

How should one think about normal subgroups intuitively? Is there any useful geometric intuition behind them? For instance, I remember reading somewhere that normal subgroups are like bundles in some ...
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59 views

Commutator Subgroup is finitely generated

It is well known that the commutator subgroup of a finitely generated nilpotent group is finitely generated, in fact all subgroups in this case are finitely generated. I am interested in infinite ...
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37 views

Group homomorphism induced by another homomorphism

I need to prove a theorem that if $\phi: G\to H$ is some homomorphism, $M\leq H$ $\phi^{-1}(M) := \{g\in G:\phi(g)\in K\}$, $M \trianglelefteq H$, then $\phi$ induces an injective homomorphism $G/\phi^...
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40 views

Proving $\gcd(n,\lvert G\rvert) = 1$

Given a finite cyclic group $G$ and any group $H$, let $\phi_1,\phi_2\colon G\to\mathrm{Aut}(H)$ be homomorphisms such that $\sigma\phi_1(G)\sigma^{-1} = \phi_2(G)$ for some $\sigma\in\mathrm{Aut}(H)$....
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85 views

Subset of a finite group

Let G be a finite group. Let $$H = \{b \in G.\ bab^{−1} \in \langle a \rangle \}.a \in G$$ Prove that if G is a finite group, then H is a subgroup of G.I think that a good approach is to prove that $$...
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168 views

Group of integer orthogonal matrices

Let $O_n(\mathbb Z)$ be a group of orthogonal matrices B st. $B*B^T=I$ with entries $b_{ij} \in \mathbb Z$. How do I show that $O_n(\mathbb Z)$ is a finite group and find its order. I need to show ...
2
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46 views

Sufficient condition to check whether a group is Noetherian

Suppose $G$ is a finitely generated group. What conditions on $G$(or some subgroups of $G$) will force it to be a Noetherian group? Of course, if all subgroups are finitely generated or ACC on ...
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86 views

Equivalent definition of abelian group

Assuming that given operation is commutative and associative, are following definitions equivalent? 1) A zero element $e_0$ exists such that $e_i+e_0=e_i$ and inverse $e_{i'}$ exists for every $e_i$ ...
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48 views

Discrete subgroups of $\mathbb R^n$ are free

Let $G$ be a nonzero subgroup of the additive group $\mathbb{R}^n$. Assume $G$ is discrete in the sense that for any $x \in G$, there exists an open set $U \subset \mathbb{R}^n$ such that $U \cap G = \...
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3answers
204 views

Are all abelian subgroups of a dihedral group cyclic?

Are all abelian subgroups of a dihedral group cyclic? Attempt: I have counter-examples for n=1,2 so I know that it isn't true for n<3. Is it true for n≥3? How do you know this?
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When is it valid to argue using my 'trivial intersection -> so many distinct elements'?

How do I know if I can make counting statements for sylow theorems that rely on intersecting trivially? I.e it is a common argument that I use in which I rely on we have $x$ amount of sylow $p$-groups ...
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43 views

What does it mean for a subgroup to fix a vector space?

I was wondering if anyone could explain what is meant by the two statements below? Any help would be greatly appreciated. Let $U=\mathbb{F}_{q}^{m}$ and $W=\mathbb{F}_{q}^{n}$ be two vector spaces of ...
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30 views

two definition in automorphism of group

Let $G$ be finite p-group and $\sigma \in Aut(G)$ (automorphism group). what does below symbols mean 1. $[G,\sigma]$ (commutator) 2. $C_G(\sigma)$ (centralizer)
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On the Commutativity of Cycles in Permutation Groups

Let $\sigma,\tau$ be two cycles in $S_n$ such that $\sigma$ and $\tau$ have different length; $\sigma$ and $\tau$ are not disjoint. Then is it always true that $\sigma\circ \tau\neq \tau\circ\...
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57 views

Extending a character of an abelian group to an overgroup

This question follows up this one, restricting the scope. Setting: Let $A$ be an abelian group and let $\chi:A\rightarrow\mathbb{C}^\times$ be a character of $A$, and let $G$ be a group containing $A$...
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54 views

what is a symmetric semigroup?

First, I think I know what is a symmetric group roughly from algebra. The group of permutation on a set with $n$ element is denoted by $S_n$, and called the symmetric group on $n$ elements (or $n$ ...
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70 views

Prove that the map $H → O$ defined by $h → hx$ is a bijection , use this result to deduce Lagrange's Theorem

Exercise: Let $H$ be a subgroup of the finite group $G$ and let $H$ act on $G$ (here $A = G$) by left multiplication. Let $x \in G$ and let $O$ be the orbit of $ x$ under the action $H$. Prove ...
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53 views

Show $D_{2n} \to GL_2(\mathbb{R})$ is an injective homomorphism

Show $\phi: D_{2n} \to GL_2(\mathbb{R})$ is an injective homomorphism, where $\phi: \textrm{rotations} \to \left( \begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{...
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92 views

Subgroups of generalized dihedral groups

A generalized dihedral group, $D(H) := H \rtimes C_2$, is the semi-direct product of an abelian group $H$ with a cyclic group of order $2$, where $C_2$ acts on $H$ by inverting elements. I know that ...
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A more formal but intuitive understanding (on a definition) of a group action

We know that a symmetric group $S_n$ acts on the set $\{1, 2,\ldots, n\}$. The definition of an action of a group $G$ on a set $S$ is a function $G\times S\to S$ such that: 1) $e\ast s=s$ 2) $g'\ast(...
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110 views

How to prove that G is a cyclic Group? [duplicate]

Suppose $G$ is a finite Abelian group and, $\forall\ n\in \mathbb{N} $, there exist at most $n$ elements in $G$ which satisfy $x^n=1$. Prove $G$ is cyclic. Thanks for your help.
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218 views

Normalizer of a Sylow 2-subgroups of dihedral groups

I can't solve the following exercise which is the last exercise in page 146 of Dummi & Foote's Abstract Algebra: Let $2n=2^ak$ where $k$ is odd. Prove that the number of Sylow 2-subgroups of $...
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1answer
53 views

Polynomial representation of elliptic curve points (Frobenius Endomorphism)

I'm trying to understand the Schoof algorithm for counting the number of points on elliptic curves in finite fields. I.e. the most basic algorithm to efficiently determine $\#E(F_p)$. For literature, ...
2
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1answer
35 views

How many subgroups are there in an elementary-$p$ group

$C_p\times C_p$ has $1+1+(p-1)=p+1$ subgroups of order $p$ (all of which $\cong C_p$), so it has $1+(1+p)+1=3+p$ subgroups totally. But how many subgroups in $C_p\times C_p \times C_p\times C_p$? (...
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Pretty easy equations of elements in a group

Problem $G$ is a group generated by $a,b\in G$ such that $a^5=e$, $aba^{-1}=b^2$ and $b\ne e$. I want to find the order of $b$. Attempt I tried to multiply the second equation from right by $a^{4}$:...
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Showing that an operation has inverses

I'm trying to show whether or not $(a, b) \cdot (c, d) = (ac-bd, ad+bc)$ on the set $\Bbb R \times \Bbb R$ with the origin deleted forms a group, an abelian group, or neither. I've shown that ...
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87 views

How to show that that not every element has an inverse

I just showed that the operation $ (a, b) \cdot (c, d) = (ac, bc+d) $ on the set $ \{(x,y) \in \Bbb R\times \Bbb R \mid x \neq 0 \} $ forms a group. I'm now looking to determine if the same operation ...
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58 views

Point conversion between Twisted Edwards and Montgomery curves

With the great help of Birational Equvalence of Twisted Edwards and Montgomery curves I know how to convert twisted Edwards curves into their birationally equivalent Montgomery counterparts where I ...
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52 views

Birational Equvalence of Twisted Edwards and Montgomery curves

I'm trying to understand the birational equivalence between Twisted Edwards and Montgomery curves and try to calculate some examples. In particular, as an example, I'm looking at the Ed25519 Twisted ...
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1answer
41 views

Show that $U(\mathbb Z_{p^n})$ is cyclic by considering the order of $1+p$

Show that $U(\mathbb Z_{p^n})$ (the group of units) is cyclic for $p$ an odd prime, $n \in \mathbb N$. We are given a hint to consider the order of $1+p$ in this group. I have no idea how this leads ...
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Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.

Let $a$ be an element of a group G. Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$. My Try: I proved the backward direction. For ...
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60 views

Is this group of matrices a $p$-group?

Let $R$ be a discrete valuation ring with the maximal ideal $\mathfrak{m}=(\pi)$ and residue field $k$ of positive characteristic $p$. Now consider $\mathrm{M}_n(\pi^iR/\pi^{i+1}R)$, $n\times n$ ...
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226 views

What finitely generated amenable groups are known to be LERF?

I know that finitely generated nilpotent groups are LERF (LERF means "subgroup separated"). I'm looking for examples (many, if possible) of groups which are: Finitely generated, but infinite ...
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1answer
45 views

$G$ be a group of order $p^n$ and $H$ be any subgroup of $G$; then does there exist $x \in G\setminus H$ , such that $xH=Hx$?

Let $G$ be a group of order $p^n$ , where $p$ is a prime and $n \in\mathbb N$ and $H$ be any subgroup of $G$; then does there exist $x \in G\setminus H$ , such that $xH=Hx$ ?
2
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Let $H$ be normal subgroup of $G$. If $G/H$ is cyclic group generated by $aH$, prove that $G=KH$ where $K=\langle a\rangle$.

Let $H$ be normal subgroup of $G$. If $G/H$ is cyclic group generated by $aH$, prove that $G=KH$ where $K=\langle a\rangle $. I would like someone to check my solution. First of i will prove that $G$ ...
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1answer
61 views

Determine the galois group of $x^5+sx^3+t$

im trying to show that the galois group of $x^5+sx^3+t$ over $\mathbb{Q(s,t)}$ is $S_5$. By just looking at the discriminant, it has to be $S_5$ or $F_{20}$. I know i could distinguish between those 2 ...
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167 views

The class equation of the octahedral group

I know that the class equation of the octahedral group is this: $$1 + 8 + 6 + 6 + 3$$ I think the $8$ stands for the $8$ vertices, the $6$ could be $6$ faces and $6$ pairs of edges. Then what is the $...
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Kernel of homomorphisms of the Baumslag-Solitar group BS(n,n)

I would like to find the kernel of the following homomorphisms and show that thoses kernels have trivial intersection. $H:=\mathbb{Z}*\mathbb{Z}/n\mathbb{Z}=\langle p,q| q^n=1\rangle$ $K=\mathbb{Z}^...
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1answer
32 views

Homomorphic image if smaller fails to exist

Suppose a finite group $G$ has no homomorphic image of order $n$. Is it possible for $G$ to have a homomorphic image of order a multiple of $n$? My gut says "no", as the larger homomorphic image ...
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2answers
42 views

Equations in groups

I want to solve an equation $$f(\sigma , \tau , \delta)=1$$ where $\sigma,\tau,\delta$ are elements from a given group $G$, and $1 \in G$ is the unit element. When I say solve I mean give sufficient ...
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1answer
118 views

Rank 3 permutation groups

Let $G \leq Sym(\Omega)$ be a finite permutation group of rank 3, $\alpha \in \Omega$ and $g,h \in G$ such that $x_1 := g(\alpha)$ and $x_2 := h(\alpha)$ are not equal. Now my question is: Is there ...
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106 views

Homomorphism of Free Groups

I am reading the theorem of homomorphism of free group from Fraleigh's text in $\S$36 and could only get a fuzzy idea at best: Let $G$ be generated by $A = \{a_i \mid i \in I \}$ and let $G'$ be ...
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1answer
81 views

Prove G is Abelian

Let $G$ be a group and $a,b\in G$. given that $(ab)^k=a^k b^k$ and $(ab)^{k+2}=a^{k+2} b^{k+2}$ for some $k\in \mathbb N$. prove that $G$ is abelian. So far my attempt was: $(ab)^{k+2}=(ab)(ab)^k(...
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Group action of $GL(2, F)$ on the projective line $P(F)$

I refer to section 8.3, page 119 of Algebra, A Computational Introduction by John Scherk. It is about group action of $GL(2, F)$ on the projective line $P(F) = F \cup \{\infty\}$. Given a matrix $\...
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$G$ finite, $P < G$ a Sylow p-subgroup, $N_{G}(P)$ the normalizer is contained in $H < G$, show $N_{G}(H)=H$

Let $G$ be a finite group and $P<G$ be a Sylow p-subgroup. Let $N_{G}(P)$ be the normalizer of $P$ in $G$. Let $H<G$ be a subgroup containing $N_{G}(P)$. Prove that $N_{G}(H)=H$. I've been ...
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1answer
33 views

Generating Finite Groups By Random Premultiplication With Generators

Let $G$ be a finite group with identity $e$ and $S$ be a set which generates $G$. Is it always possible to define a procedure of the form: Start with $x=e$. With probability $p_1$, replace $x$ with $...
2
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1answer
72 views

Showing that the product $x*y := \frac{x+y}{xy+1}$ is a group operation on $(-1, 1)$ [duplicate]

I need to show that the following is an abelian group: $$x*y = \frac{x+y}{xy+1}$$ on the set $\{x \in \Bbb R \,|\, -1 < x < 1\}$. I have been working on this problem, trying to show ...