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

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Functions $h$ such that $h(x*x') = f(x) * g(x').$

Definition 0. Call a magma $X$ surjective iff the distinguished binary operation of $X$ induces a surjective function $X \times X \rightarrow X$. Now for the main idea: Definition 1. Let: ...
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Herstein - Topics in Algebra - Polynomial rings page 157

In Chapter 3.9 of his book "Topics in Algebra" , 2nd ed, Herstein describes an example of a Quotient ring, namely $ F[x]/(x^3-2) = F[x]/A $ where $F = Q $ the rationals, and $(x^3-2) = A $ is the ...
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Determining order of an element in a symmetric group

Specifically, how is an element in $S_5$ e.g $(1 2 3) (4 5)$ have order $6$? Can someone explain this?
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Why $\mathbb Z/ 2 \mathbb Z$ is not a free module?

I am reading some abstract algebra notes about free modules. It says that not all modules are free and the example to illustrate this is $\mathbb Z/ 2\mathbb Z$ (as a $\mathbb Z$-module) is not a free ...
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Show $G$ is a group, when there doesn't seem to be an inverse?

I would like to show that $G=\left\{\begin{bmatrix} a&a\\a&a \end{bmatrix}\mid a\in\mathbb{R}\setminus\{0\}\right\}$ is a group under matrix multiplication. I've already verified that ...
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Show that the intersections of the $G_s$ is normal subgroup of $G$

I need to prove that given a group $G$ acting in a set $S$, the intersection of the stabilizers $G_s$, where $G_s:=\{g\in G: g.s=s\}$ and $s$ varies through all $S$, is a normal subgroup of $G$. But ...
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Epimorphism from G to Z

I've got a problem with this exercise, I'd be thankful if someone could help. Let $G$ be a group and let $f$ be an epimorphism from $G$ to $\mathbb{Z}$. Show that for every positive integer $n$, $G$ ...
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36 views

Kernel of cononical ring homomorphism

My question is what exactly does the kernel of a map $\phi:R\to R/S$ look like? By definition my first guess it write $\mathrm{ker}\phi=\{a:\phi(a)=0\in R/S\}$ but the "zero element" in $R/S$ is just ...
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Combinatorial group theory books

I would please like some recommendations for an introductory level book on combinatorial group theory, by which I mean a group theory book which places emphasis on generators and relations and free ...
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Natural isomorphisms of the forgetful functor

Let $U: \mathbf{Groups} \rightarrow \mathbf{Sets}$ be the forgetful functor. Must every natural transformation $\eta: U \rightarrow U$ be a natural isomorphism?
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On a Proof in Galois Theory

We have the following lemma (used in the proof of Abel's Theorem): If $\text{Car}(F)=0$, $E/F$ is a radical extension, and $K/E$ is the Galois closure of $E/F$, then $K/F$ is also a radical ...
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The rationals as a direct summand of the reals

The rationals $\mathbb{Q}$ are an abelian group under addition and thus can be viewed as a $\mathbb{Z}$-module. In particular they are an injective $\mathbb{Z}$-module. The wiki page on injective ...
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Rings with two elements.

Let $K$ be an assotiative ring with 1 nonzero multiplication. Is it true that if $K$ consists of two elements then $K \cong \mathbb{Z}_2?$ It is clear that second element is $0$ and $1 \cdot 1=1, ...
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Are the quaternions obsolete in pure mathematics?

I remember I read an article saying that "The quaternions $\Bbb{H}$ are obsolete in pure mathematics since the theory of vectors has been developed enough, however it is useful in computer science". ...
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Let G be a finite cyclic group of order n. If d is a positive divisor of n , prove that x^d = e has exactly d distinct solutions in G

well i know that for a group to be cyclic then there must exist an element in G for example we call it g such that $G = \langle g\rangle$ and so $g^0 = e$ and $g^0 = g^n = e$ hence ...
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List the elements of $\langle\frac{1}{2}\rangle$ in $(\mathbb{Q},+)$ and in $(\mathbb{Q}^*,\times)$.

List the elements of $\langle\frac{1}{2}\rangle$ in $(\mathbb{Q},+)$ and in $(\mathbb{Q}^*,\times)$. where $\mathbb{Q}^*:=\mathbb{Q}\setminus\{0\}$ My attempt: Well, I know that $\langle ...
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Morphism of $k$-algebras between abelian group of $n \times n$ matrices and $m \times m$ matrices

Problem Let $k$ be a field and $f:M_n(k) \to M_m(k)$ be a morphism of $k$-algebras ($n,m \in \mathbb N$). Prove that $n$ divides $m$. I have no idea what to do here, I thought that maybe I should ...
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Show that there exists a prime ideal $P$ of $R$ such that $ I \subseteq P$ and $P \cap S=\emptyset$ [duplicate]

Show that there exists a prime ideal $P$ of $R$ such that $ I \subseteq P$ and $P \cap S=\emptyset$, if $S$ is multiplicatively closed subset of $R$, $I$ is an ideal of $R$ such that $I \cap S = ...
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Showing finite lattices are isomorphic to their sets of ideals.

I'm working out a problem from Gratzer '71. We are to show that any finite lattice $L$ $\simeq$ $I(L)$. My attempt was as follow: prove the contrapositive. So assume $L$ is not isomorphic to $I(L)$. ...
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A question about Klaus Hulek algebraic geometry

I'm reading Klaus Hulek's algebraic geoemtry and there is something that I can't understand. Here it says that if {p,q} is a counterexample with minimum max{deg p , deg q}, then it can be assumed ...
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Can $\mathbb{Z}$ be endowed with operations that give it the structure of a field?

Does there exist some definition of addition and multiplication for which the set of all integers is a field?
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Find a group $G$ and $H\subseteq G$ that shows $(H\leq G$ iff $ab\in H)$ is not valid if $G$ is infinite.

The theorem says, let $G$ be a finite group with $H\subseteq G$, with $H\neq \varnothing$. $H \leq G$ iff $ab \in H$ for all $a, b \in H.$ I have no idea how to start.
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Definition of Tensor Product of Modules

I am really struggling to understand several parts of the definition of tensor product given in my lecture notes: Definition of the tensor product *Denote by L the free A-module with a basis ...
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Uniqueness of Duals in a Monoidal Category

Given a monoidal category ${\cal C}$, and $X \in {\cal C}$, we define a left dual of $X$ to be an object $X^*$ such that there exist morphisms $\epsilon:X^* \otimes X \to I$, and $\eta:I \to X \otimes ...
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Prove that the kernel of a group homomorphism $\phi$ is a subgroup and that $\phi$ is injective

I am solving the following exercise: Let $\phi : G_1 \rightarrow G_2$ be a homomorphism (where $G_1$ and $G_2$ are groups) and $\ker \phi := \{ g \in G_1 \mid \phi(g) = e \}$ now I have to ...
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Give an example of a maximal ideal in a noncommutative ring which is not prime

While trying to find an example, I came up with this: Since if $J$ is an ideal of a the ring $M_n(R)$, where $R$ is a commutative ring, then $J=M_n(I)$ for some ideal $I$ of $R$. IF I could show that ...
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Question Regarding a Group G

Let $G$ be a group and let $a,b \ \epsilon \ G.$ Show that $(a * b) * (a' * b') = e$ if and only if $(a * b)$ = $(b * a)$ Note that * is a binary operation, $a'$ and $b'$ are inverses of $a$ and ...
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Determining if an extension is a normal extension [MORANDI'S BOOK]

I'm dealing with the following problem: Let $K$ be a field and suppose that $\sigma\in\mbox{Aut}(K)$ has infinite order. Let $F$ be the fixed field of $\sigma$. If $K/F$ is algebraic, show that ...
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Homomorphism $\phi: \mathbb{Z} \times \mathbb{Z} \rightarrow G$ proof

Let $G$ be a group. Let $h,k \in G$ and let $\phi:\mathbb{Z}\times \mathbb{Z}\rightarrow G$ be defined by $\phi(m,n)=h^mk^n$. Give a necessary and sufficient condition, involving $h$ and $k$, for ...
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How to find out if two cycles are conjugate?

Let for example $a=(14395)(26)(78)$ and $b = (154)(2368)(79)$ be elements of $S_9$. I know that by definition, conjugate elements of a group $G$ are elements $x,y \in G$ such that $x=aya^{-1}$ for ...
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Product of finite order elements in a group

Let $G$ be a group. Let $a,b\in G$ be of finite order. Prove or disprove: (1) If $ab$ has finite order, then $ba$ has finite order. (2) If $ab$ has finite order, then $a^{-1}b^{-1}$ has finite ...
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Commutator of a group

A commutator in a group $G$ is an element of the form $ghg^{-1}h^{-1}$ for some $g,h\in G$. Let $G$ be a group and $H\leq G$ a subgroup that contains every commutator. $(a)$ Prove that $H$ is a ...
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What's the best way to define $\mathbb{H}$?

I had once defined $\mathbb{C}$ as $\mathbb{R}\times\mathbb{R}$ as a set, then I found it extremely uncomfortable when doing complex analysis or measure theory. So I changed its definition so that ...
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Suppose that $K$ is a field and that $f$ and $g$ are relatively prime in $K[x]$. Show that $f - Yg$ is irreducible in $K(y)[x]$.

I'm a bit confused of the notation $K(y)[x]$, is that simply $K[y][x]$ so... $K[y,x]?$ Anyways, here's my attempt at trying this before I get stuck. Since $f$ and $g$ are relatively prime, that ...
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Zerodivisors in polynomial rings over a non-commutative ring

Prove that if $f \in R[x]$ is a zero divisor then $\exists r(\neq 0) \in R$ s.t $rf=0$, where $R$ is a ring. I know that for $(a_0+a_1x+ \cdots ...
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Finite Abelian Group Proof

Show that a finite abelian group is not cyclic if and only if it contains a subgroup isomorphic to $\mathbb{Z}_p \times \mathbb{Z}_p$ for some prime $p$. I'm not sure what to do. Any proofs or hints ...
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An alternative proof for the units of $U_q(\mathfrak{sl}_2)$ using Ore extensions.

I would like to establish what the set of units are in the quantized enveloping algebra $U_q(\mathfrak{sl}_2)$. First, I recall the definition of the quantized enveloping algebra- throughout the ...
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Calculating the Order of An Element in A Group

First of all, I am very new to group theory. The order of an element $g$ of a group $G$ is the smallest positive integer $n: g^n=e$, the identity element. I understand how to find the order of an ...
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Is $\alpha$ conjugate to $\beta$?

Let $\alpha=\left(\begin{array}{ccccc} 1&2&3&4&5\\ 2&1&4&5&3 \end{array}\right)$ then is $\alpha$ conjugate to $\beta=\left(\begin{array}{ccccc} ...
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Let $L \supset K$ be an extension field. Prove $K(a_0, a_1, \ldots, a_n) = K(a_0, a_1, \ldots, a_{n-1})(a_n)$ ? Why smallest subfield with properties?

Suppose $L \supset K$ is an extension field. How do one prove rigorously that $K(a_0, a_1, \ldots, a_n) = K(a_0, a_1, \ldots, a_{n-1})(a_n)$ ? I understand that $K(a_0, a_1, \ldots, a_n)$ is the ...
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How can we find an element of largest order in $S_n$ in general? [duplicate]

How can we find an element of largest order in $S_n$ in general? For the small orders we can find by trial and error method. For example in $S_3$: 6 is the largest possible order. Similarly for ...
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What is the definition of finitely generated?

To be specific, here is an example. Note that $(\mathbb{Z},+)$ is a group. Definition 1. (Lang) Let $G$ be a group and $S\subset G$. If $G$ is the intersection of all subgroups $H$ of ...
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What would be an example that rank of a subgroup of a free group is greater than the rank of free group?

Let $G$ be a free group and $H$ be a subgroup of $G$. Then, $H$ is also a free group. Let $R_G,R_H$ be ranks of $G,H$ respectively. what would be an example that $R_H>R_G$?
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Divisibility of polynomials in a subfield of a field.

I am trying to prove the following assertion: Let $K\subset L$ be fields, let $f,g\in K[x]$ be such that $f\mid g $ in $L[x]$, then $f\mid g$ in $K[x]$. We clearly have that $fh=g$ for some ...
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Let $H\subset G, K\subset H$ be normal subgroups such that $K$ is normal in $G$ .Describe $(G/K)/(H/K).$

First of all I would like to say hello to all of you! I am new here! :-) I am very pleased to have found this site. So I can help others if they need me and get help when I need it, really a great ...
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Definition of direct sum of modules?

I have just started studying modules, and am trying to figure out the definition of the direct sum of modules but I'm having trouble since different sources seem to give different definitions, for ...
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How to show that there is a bijective correspondence between two sets of prime ideals

I'm trying to solve this Algebra Problem, and I'm not quite sure, if I'm on the right way. Let $R$ be a commutative ring and $S \subset R$ a multiplicative subset. Show that $p \to pS^{-1}R$ ...
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Is every free subgroup a direct summand?

Let $G$ be an abelian group. Suppose that $F$ is a subgroup of $G$ such that $F$ is free. Does there necessarily exist a subgroup $H\subset G$ such that $G\cong F\oplus H$? Motivation: In Lang's ...
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Is every symmetric group generated by some cycles?

Actually, I have two questions here: How to show that $D_3$, the dihedral group, is isomorphic to $S_3$ in details? Is every symmetric group generated by some cycles? For question (1), I know ...