Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, among other topics.

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Centralizer, normalizer, and center of a dihedral group

Let $A := \{1, r, r^2,..., r^{n-1}\}$. Compute $C_{D_{2n}}(A), N_{D_{2n}}(A),$ and $Z(D_{2n})$. So far I figured that all of the rotations are in the centralizer/normalizer, because all rotations ...
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Set of all permutations on n generating function

Show that $S_n = \langle (1\ 2), (1\ 2\ \ldots\ n) \rangle$ for all $n \geq 2$. I'm not sure how to approach this one.
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2answers
32 views

Prove that $\mu:G\times G \rightarrow G$ is a homomorphism if and only if $G$ is abelian.

Given $\mu:G\times G$ be the operation on a group $G$; that is, $\mu (a,b)=ab$. Prove that $\mu$ is a homomorphism if and only if $G$ is abelian. I have no problem on proving the necessary ...
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1answer
15 views

Show that $m\mathbb{Z}/md\mathbb{Z} \cong \mathbb{Z}/d\mathbb{Z}$ if and only if $(d,m)=1$

Show that $m\mathbb{Z}/md\mathbb{Z} \cong \mathbb{Z}/d\mathbb{Z}$ as rings without identity if and only if $(d,m)=1$. I know that if $\phi : A \to B$ is a epimorphism ring and $A$ is a unit ...
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1answer
17 views

Compute the matrix with the standard basis.

Let $X$ be the left representation of $S_3$. Compute the matrix $X(132)$ in the standard basis $S = {\{123,132,213,231,312,321}\}$. attempt: In general if $G = S_3 = {\{\pi_1, \pi_2,..,\pi_6}\}$, ...
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4answers
29 views

Show that $\ker(\phi)$ is a maximal ideal if and only if $B$ is a field

Let $A$ and $B$ two commutative rings with unity $1_A \not= 0_A$ and $1_B \not= 0_B$. Consider $\phi : A \to B$ a ring epimorphism. Show that if $\ker(\phi)$ is a maximal ideal, $B$ is a field. I ...
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1answer
12 views

Let $n \geq 2$ be an integer, and consider the group $Z_n:=(\{0,1,. . .,n-1\}, +_n)$. Let $k \in Z_n$ \ $\{0\}$.

Let $n \geq 2$ be an integer, and consider the group $Z_n:=(\{0,1,. . .,n-1\}, +_n)$. Let $k \in Z_n$ \ $\{0\}$. Show that the following statements are equivalent: (a) $\gcd(n,k)=1$, (b) the only ...
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0answers
13 views

Simplifying a coset

Let G be a group and let $M,N \leq G$ be normal such that $G = MN$. Prove that $G/(M \cap N) \cong (G/M) \times (G/N)$ I have found a solution to this question here: ...
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1answer
26 views

First isomorphism theorem application

Let G be a group with, $N\subset G$ a normal subgroup, And assume that $H$ is a subgroup of $G$, $H\subset G$. Further $HN=G$ and $H\cap N = \{e\}$ . Prove that $H$ generates the cosets of $N$ in ...
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How to find identity element of a set (under modular) operation?

Question 1) Can the set of $\{0, 1, 2, 3\}$ under the operation of modulo-$4$ addition and multiplication form a group as well as a field ? If yes then how and if not then why ? Question 2) How to ...
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15 views

Regular semigroups- normal semigroups!

If $S$ is a Clifford semigroup with the set of idempotents $E$, then $S'$ be a sub-semigroup of $S$ ( so $S$ be a semilattice with the same set of idempotents( $E$) such that for every $e \in E$, ...
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0answers
25 views

Unramified morphism

I was reading the following page: https://ayoucis.wordpress.com/2014/04/06/unramified-morphisms/ and there are several things I do not understand and would like to clarify. First doubt The ...
2
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1answer
31 views

Can this extension of fields be transcendental?

Let $(R, \mathfrak m)$ be a local integral domain which is contained in a field $K$. Let $0 \neq x \in K$ be such that $\mathfrak m R[x]$ is a proper ideal of $R[x]$ (one can show for any $x$ that ...
2
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1answer
24 views

Finite abelian group as Z-module

If $M$ is a finite abelian group then $M$ is naturally a $Z$-module. Can this action be extended to make $M$ into a $Q$-module ?
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20 views

What does it mean a transitive permutation?

let $X$ be a finite set. Let G be a group. What is the meaning of $G$ is a transitive permutation on set $X$?
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32 views

Group theory: Intuition as to what a group is [duplicate]

In group theory the group is an algebraic structure consisting of a set which has elements associated with definite finitiary operations. Can an intuitive explanation be provided as to what this ...
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0answers
9 views

Representation of left ideals of the matrix ring [duplicate]

We know that $J$ is an ideal of the full matrix ring $S=M_n(R)$ if and only if $J$ is the ring of all $n\times n$ matrices over $I$ for some ideal $I$ of the ring $R$ with identity. Now, my question ...
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1answer
33 views

Doubt about associative property of a group (Abstract Algebra). [on hold]

I am new to abstract algebra and I have a doubt about the associative property. Suppose a set is given, such as $G=\{0,1,2,3,4\}$ under $\pmod{5}$ addition operation and we have to check whether $G$ ...
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1answer
21 views

Prove $(H \times 1)(1\times K)= H\times K$ where $H,K$ are groups.

Prove $(H \times 1)(1\times K)= H\times K$ where $H,K$ are groups. Suppose $x=ab,a\in H\times 1,b\in 1\times K$ Then $x=(h,1)(1,k)$ where $h\in H,k\in K$ Hence $x=(h,k)\in H\times K$ Let ...
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1answer
20 views

Classifying groups of order $6$ using semidirect products

Let G be a group of order 6. I am able to do the exercise without semidirect products($G \cong Z_6 $ or $S_3$) but I don't know how to use semidirect products to do this. By Sylow's theorem, there ...
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1answer
45 views

Why do polynomials and integers both have a long division algorithm?

The grade-school long division algorithm and the polynomial long division algorithm are identical, if I'm not mistaken. Why is this the case? Are the two algebraic structures identical in some sense? ...
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40 views

Spliting subspaces and fields

I'm sure that the following is true, but I can't prove it. Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a tower of finite fields and $A = \{\theta\in K: ...
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2answers
39 views

$2\otimes 1$ is non-zero in $2\mathbb Z\otimes_{\mathbb Z} \mathbb Z/2\mathbb Z$.

I had the following doubt: Show that the element $2\otimes 1$ is $0$ in $\mathbb Z\otimes_{\mathbb Z} \mathbb Z/2\mathbb Z$ but not a zero in $2\mathbb Z\otimes_{\mathbb Z} \mathbb Z/2\mathbb Z$. ...
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3answers
29 views

Prove that the four roots of unity form an abelian multiplicative group

My question is: Let $i = \sqrt{-1}$. Prove that the four roots of unity $\{1, -1, i, -i\}$ form an abelian multiplicative group. I know that abelian group is a group with commutative property. ...
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1answer
31 views

Order of a subgroup generated by two permutations

Let $$\alpha=(1,3,12,7)(8,5,6,2,11)(4,9,10)$$ $$\beta = (1,5)(6,8,11)(12,3,2,7)(4,9,10).$$ How does one prove that a subgroup G that contains $\alpha$ and $\beta$ has order $o \ge 120$? ...
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0answers
33 views

Permutations of $S_n$ whose order divides a positive integer $m$

For which $n,m \in \mathbb{N}$ is $$K_m = \{\sigma \in S_{n}: \text{ord}(\sigma) \text{ divides } m \}$$ a subgroup of $S_{n}$? How does one approach such a problem?
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Adjoint pair of functors and cogenerator elements

Let $F:\mathcal{A} \rightarrow \mathcal{B}$ and $G:\mathcal{B} \rightarrow \mathcal{A}$ be additive functors between abelian categories, such that $(F,G)$ is an adjoint pair. If $B \in \mathcal{B}$ is ...
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2answers
41 views

Is $(\mathbb{Z}_4, +_4)$ isomorphic to $(\langle i\rangle, *)$

Now, I am not sure, but I think that the second group is cyclic, because of the way it's defined $(\langle i\rangle,*)$. $i$ is probably the generator of the group. But, how can I prove that ...
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1answer
26 views

A group $G$ is finitely generated iff if there is a surjective homomorphism $F(\{1,…,n \}) \to G$

This is taken for granted in Algebra: Chapter 0 by Paolo Aluffi. Here is a definition of subgroup generated by a subset from the book: Let $A \subseteq G$. We have a ujnique group homomorphism ...
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1answer
50 views

Find error in abstract algebra proof

I suspect that the proof below is flawed. I did not use the hypothesis "$\ker(h) \subseteq \ker(k)$" when proving sufficiency. Lemma. $ $ Let $G$, $H$, $K$ be groups, let $h : G \to H$ and $k : G ...
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1answer
32 views

Suppose that $f : C → C$ is an isometry such that $f(0) = 0$, $f(1) = 1$ and $f(i) = −i$. Prove that $f(z) = \bar z$ for all $z ∈ C$.

Suppose that $f : C → C$ is an isometry such that $f(0) = 0$, $f(1) = 1$ and $f(i) = −i$. Prove that $f(z) = \bar z$ for all $z ∈ C$. I already have a proof for this but I would like an explanation ...
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1answer
29 views

Describing the clopen sets of a profinite group

I've read somewhere that all clopen subsets of a profinite group $$G \simeq \varprojlim\left(G_i, f_{ij}:G_i \to G_j\right)_{i,j \in I}$$ are exactly the preimages of subsets of the $G_i$'s. It's easy ...
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2answers
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Show that the set of all principal ideals is an equivalence class of the relation $\sim$

Let $A$ a integral domain and let $\mho(A)$ the set of all non-zero ideals. Show that the set of all principal ideals is an equivalence class of the relation $\sim$ that we can noted by $[A]$. ...
4
votes
1answer
51 views

Norm on “tower” of fields (the question comes from algebraic number theory)

Consider the field extensions $L\supset K'\supset K$, where both $L|K$ and $K'|K$ are finite and Galois. I want to prove that $$N_{L|K}(L^\ast)\subseteq N_{L|K'}(L^{\ast})$$ Maybe it is very easy, ...
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1answer
26 views

What are the $GL_n(F)$-orbits of a group action on the set of idempotent matrices?

Let $S= \{A \in M_{n \times n}(F):A^2=A\}$ (set of idempotent matrices). The general linear group $G=GL_n(F)$ acts on $S$ by $A.g=gAg^{-1}$ (conjugation). I'm having trouble visualizing the ...
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2answers
47 views

Find $\phi, \psi: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, each nilpotent of order 2, whose composition is idempotent

$\phi: V \rightarrow V$ is nilpotent of order 2 if $\phi \phi$ is the zero endomorphism. Now composition of two such endomorphisms need not be nilpotent of order 2. Find $\phi, \psi: \mathbb{R}^2 ...
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0answers
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Isomorphism from $\mathbb{Q}(\sqrt{2})$ to $\mathbb{Q}[x]/\langle x^2 - 2\rangle$ [on hold]

I am just now beginning my first course in Fields. Sometimes I learn best by just being absolutely certain of some basic facts. This is why I like to ask simple True/False questions that I think are ...
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1answer
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Subrings of the product of rings of algebraic integers

Let $K,L$ be a number fields and $\mathcal{O}_K,\mathcal{O}_L$ their rings of algebraic integers, and $n_K=[K:\mathbb Q]$, $n_L=[L:\mathbb Q]$. Suppose that $R$ is a subring of ...
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1answer
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I do not understand the definition of a “K-automorphism”

I am reading Ian Stewart's Galois Theory and frankly, the following definition is puzzling me Let $L:K$ be a field extension, so that $K$ is a subfield of the subfield $L$ of $\mathbb{C}$. A ...
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1answer
47 views

Show that $\mid$G$\mid$ is prime

Suppose Group G has precisely two subgroups. Show that $\mid$G$\mid$ is prime. $\mid$G$\mid$ denoted to the order of the group G. If I let $g\leq$ G and $k\leq$ G. I know that $G^p$= e where e is ...
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1answer
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Prove closed disc $D^n$ is homeomorphic to the cone $CS^{n-1}$

I need to find a continuous surjective map from $D^n$ to $CS^{n-1}$. For 2 dimensions, we can use $$f: S^1 \times I /S^{1} \times \{1\} \rightarrow D^2$$ with $f(\theta,t) = (1-t)e^{i \theta}$ ...
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1answer
30 views

If $G/K$ is isomorphic to $H$ then is $G$ isomorphic to $KH$?

I don't know if the question is trivial but I would really appreciate if someone could prove/disprove this question. Thank you in advance Edit: I meant this: Suppose that there is homomorphism $f$ ...
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What kind of algebraic structure is $\left( \mathbb{R}_{\geq 0},+,\cdot \right)$?

Let $\left( \mathbb{R}_{\geq 0},+,\cdot \right)$ denote the non-negative real numbers with usual addition and usual multiplication. Obviously, this is not a field, because $0$ is the only additively ...
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1answer
12 views

Calculating $| \langle \cup_{i=1}^n P_i \rangle |$ where $P_i$ are Sylow subgroups of G

I'm trying to prove: Let $\lbrace P_i: i \in I \rbrace$ be a set of Sylow subgroups of a finite group G, one for each prime divisor of $|G|$. Then $\langle \cup_{i \in I} P_i \rangle = G$. (From ...
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1answer
56 views

Define $\phi:\mathbb{R}^3 \rightarrow \mathbb{R}$ by $\phi(e_1) = 1$, $\phi(e_2) = 2$, $\phi(e_3)=-1$. Determine ker$\phi$ and im$\phi$

Let {$e_1,e_2,e_3$} be the standard basis for $\mathbb{R}^3$ and define $\phi:\mathbb{R}^3 \rightarrow \mathbb{R}$ by $\phi(e_1) = 1$, $\phi(e_2) = 2$, $\phi(e_3)=-1$. Determine the subspaces ...
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2answers
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CG-modules: what does this notation mean?

I am trying to solve a question, but I do not know what the notation used means. If anyone could help me out that'd be great! I don't need help doing the proof, just what the notation means would be ...
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votes
1answer
50 views

$\mathbb C$-dimension of vector space $\mathbb C\otimes_{\mathbb R}\mathbb C$

Let $\mathbb R$ be the field of real numbers, $\mathbb C$ be the field of complex numbers. Consider $\mathbb C\otimes_{\mathbb R}\mathbb C$ as a $\mathbb C$-vector space via $a(b\otimes c) := ab ...
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0answers
54 views

Prove that $u$ is a trivial unit

Let $G$ be a finite group. I want to prove that if $u\in \mathbb{Z}G$ be a torsion unit (i.e. for some $n$, $u^n=1$) of integral group ring $\mathbb{Z}G$ such that $u$ normalizes $G$, then $u$ is a ...
0
votes
2answers
32 views

Proving that rotation is an isometry in the complex plane

Consider the rotation $ρ_θ : \Bbb C → \Bbb C$ about the origin with angle $θ$ in counterclockwise direction; this can be described by the map $ρ_θ(z) = e ^{iθ} z$. Prove that $ρ_θ$ is an isometry of ...
0
votes
1answer
57 views

Show that $P_n$ is an $(n+1)$-dimensional subspace of the vector space of all real polynomials [duplicate]

Show that $P_n$ = {polynomials with real coefficients of degree $\leq$ n} is an ($n+1$)-dimensional subspace of the infinite-dimensional vector space of all real polynomials I know that $P_n$ is ...