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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Conditions for subring.

Given a ring $R$. Suppose that $S \subset R$ and $0,1 \in S$. It is also given that $a + b \in S$ and $a \bullet b \in S$ for all $a,b \in S$.Is this sufficient in showing that $S$ is a subring of ...
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Free objects in the category of dg modules

Suppose that $A$ is a dg algebra, does the category of dg modules over $A$ where morphisms are degree zero maps that commute with differential have a free object ( in general)? I have been reading a ...
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Intersection point between a line and plane: what's wrong with my calculation?

I'm trying to calculate the intersection point between a line and a plane, but apparently there is something wrong with my calculation and I don't know what exactly. The exercise goes as follows: ...
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Basis of $\mathbf{Q}[x]$

I wanna show that the binomials $\binom{x}{k}$ for $k=0,1,\ldots$ form a basis of the $\mathbf{Q}$-vector space $V=\mathbf{Q}[x]$. I can show that for fixed $m\in\mathbf{N}$ the $\binom{x}{k}$ ...
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29 views

Index of center $Z(G)$ is finite implies the number of elements of conjugacy class is finite

Exercise Let $G$ be a group such that its center $Z(G)$ has finite index in $G$. Show that every conjugation class has finite elements. I don't know how to attack the problem. I thought the ...
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Algebra: Does every group whose order is a power of a prime p contain an element of order p?

I need to solve this using Lagrange's theorem and simple corollaries. Thanks.
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Field isomorphism and order of elements

I know that group isomorphism preserves order of element but can someone plese tell me does field isomorphism preserves order of elements?
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55 views

Commutative ring and maximal ideal problem

Let $A$ be a commutative ring and $M$ be a proper maximal ideal in $A$. Show the following properties: (a) If each $a \in A \setminus M$ is a unit element in $A$, then $M$ is the only maximal ideal ...
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How to find identity and inverse for the group $(\mathbb{Z}, \ast)$, where $a \ast b = a+b-ab$

I'm trying to figure out how to find identity and inverse for the group defined as $(\mathbb{Z}, \ast)$, where $a \ast b = a+b-ab$? I've found that closure and the associative property are both true, ...
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35 views

Hom and Tensor product

Let $\gamma:V\to W\otimes L$, $\operatorname{rank}(L)=1$, be a map. Using $\operatorname{Hom}(V,W\otimes L)\cong \operatorname{Hom}(V,W)\otimes L$ we can get $\gamma':V\to W$. If $\gamma'$ not be an ...
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Reverse Polish notation in (abstract) algebra

If I have something like $\phi\circ \psi(x)$ this means first apply $\psi$ and then $\phi$. Going right to left is pretty contrary to my intuition. In computer science some programming languages (and ...
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54 views

Question concerning a sequence in GAP

I would like to know, what's the best (fastest) way to programm the following in GAP (perhaps using some functionality from the QPA package): Input: $n\geq 2$ Output: A list of all sequences ...
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Find the units of the ring $\mathbb{Z}_6[x]/\langle 2x+4 \rangle$. [on hold]

Find the units of the ring $\mathbb{Z}_6[x]/\langle 2x+4 \rangle$.
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29 views

Proving a submodule is Noetherian

I'm doing some independent study with a professor in Ring/Field theory (I'm an undergraduate) and I have been having a hell of a time wrapping my head around problems involving Noetherian rings and ...
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28 views

If f(x+a) is irreducible over F then f(x) is irreducible over F

Actually we can prove the statement in the reverse way, Here a is non zero element, F is a field Suppose f(x) is reducible then f(x) = g(x)h(x), then f(x+a)= g(x+a) h(x+a). But I can't understand ...
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Group theory and group representation

I am fairly new to group theory and representation. I am currently looking at faithful representations. I am not quite sure what is the "use" of a faithful representation. I cannot find any "easy to ...
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Abelian group action exercise

Let $X$ be a set with $n$ elements and let $G$ be an abelian group acting on $X$ such that: $$(i) \space gx=x \space \forall x \implies g=1,$$ $$(ii) \space \forall x,y \in X, \exists g: gx=y.$$ Show ...
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Existence of finite indexed normal subgroup for a given finite indexed subgroup.

Prove that if H has finite index n then there is a normal subgroup N of G with $N \subset H $ and [G:N]$ \le $n!. I tried to solve the problem but could not done exactly. Since [G:H]=n , Let A={$a_i ...
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47 views

A property of minimal prime ideal

Let $R$ commutative ring with unity, $S\subseteq R$ subring, $p$ minimal prime ideal of $S$. Show there exists a minimal prime ideal $q$ in $R$ with the property that the contraction $q^c=q\cap S=p$. ...
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28 views

The root of $x^2+[1]=[0]$ in $\mathbb{Z}_p$

In $\mathbb{Z}_p$, where $p$ is a prime, how many roots of $x^2+[1]=[0]$? It is equivalent to show $[x^2]=[p-1]$,when p=3,there is non. When p=5, $x=2$,does there exist any rule of it
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Why the kernel is a normal subgroupl?

Let $G$ be a group acting on a nonempty set $A$. Why the kernel of the action is a normal subgroup of $G$?
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Zero object equivalent assertion

Let C be a category with zero object $0$. (i) Prove that for $A \in C$ the following assertions are equivalent: (a) A is a zero object; (b) $id_A$ is a zero morphism; (c) there is a monomorphism ...
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Mapping set of integers to irrational numbers.

Mapping Integers to Irrationals..maybe even primes? Hi i'm an undergrad currently working on a research project. I recently thought of a question that I believe would help me greatly,but have ...
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How do I construct the multiplication of a quotient group?

The question is: If $G$ is the group of all nonzero real numbers under multiplication and $N$ is the subgroup of all positive real numbers, write out $G/N$ by exhibiting the cosets of $N$ in $G$, ...
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23 views

How to prove that the evaluation map is a ring homomorphism?

This is a really easy question, but I'm stuck in the logic of it... Let $F$ be an integral domain and $F[x]$ its polynomial ring. Let $a\in F$ fixed, define $\phi: F[x]\to F$ as ...
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local dimension of irreducible varieties

If $X$ is an irreducible (quasi-)affine variety it is well known that each maximal sequence $C_0 \subsetneq C_1 \subsetneq \dots \subsetneq C_d$ of irreducible closed subsets has the same length $d = ...
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There are no simple groups of order $27p$

Statement Show that there are no simple groups of order $27p$ for any $p$ prime number. I got stuck with this problem, I'll write what I've done so far: Suppose $G$ is a group with $|G|=3^3p$, $p$ ...
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Abstract algebra: equivalence relations on vector spaces [on hold]

Let $W$ be a subspace of a vector space $V$ over $\mathbb{R}$. (that is the scalars are assumed to be real numbers ). We say that two vectors $u,v \in V$ are congruent modulo $W$ if $(u-v)\in W$, ...
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Linear Map Operation Notation

I'm trying to determine the notation for a linear map operation that my teacher is using. One particular example is in the homework problem I am doing, in which T∈ L(U,V) and S ∈ L(V,W). I am asked to ...
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Give an example of three different groups with eight elements

Give an example of three different groups with eight elements. Why are the groups different? One particular answer that I found was the groups $\mathbb{Z}_8$, $\mathbb{Z}_4 \times\mathbb{Z}_2$, and ...
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Abelian Kummer Extension

A field extension of the form $\mathbb{Q}(\zeta_n, \sqrt[n]{\beta})$ where $\zeta_n$ is a primitive $n$th root of unity and $\beta \in \mathbb{Q}(\zeta)$ is called a Kummer extension. Even though ...
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Prove $\mathbb{Q}(\sqrt2,\sqrt2i)=\mathbb{Q}(\sqrt2+\sqrt2i)$

I want to know why the following two are equivalent: $$\mathbb{Q}(\sqrt2,\sqrt2i)=\mathbb{Q}(\sqrt2+\sqrt2i)$$, where $\mathbb{Q}$ is the rational number field, and ...
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Irreducible permutations in $S_n$

Let $n$ be an integer $\geq 2$ , let $\tau \in S_n$ and let $X$ be a nonempty subset of $\{1,2,\ldots,n\}$. Say that $\tau$ fixes $X$ if $\tau(X)=X$, and say that $\tau$ acts irreducibly provided the ...
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Commuting between Symmetric Groups

I had a homework question that asked to give an example of a subgroup of $S_4$ that is not normal. I found such an example but I'm not sure why it worked. If I understand the normalizer correctly ...
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How many different groups? [on hold]

My question is how many groups can we have with just 2, 3 or 4 elements? Am I looking at subgroups?
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24 views

How many distinct prime ideals in $\mathbb Z/p^2q\mathbb Z$

How many distinct prime ideals are there in $\mathbb Z/p^2q\mathbb Z$ , where $p,q$ are distinct primes?
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Nonabelian group of six elements

What is an example of a six-element group that is not abelian? I can't think of any. It is very possible that I am overthinking this. Thank you for any help.
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Projective Resolutions of Finite Type.

Definition. An $R$-module $M$ is said to be of type $FP_n$ if there is a projective resolution $\mathscr{P}$ of $A$ with $P_i$ finitely generated for all $i\leq n$. If the modules $P_i$ are finitely ...
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Proving a defined group $(G,*)$ is isomorphic to $(\mathbb{R},+)$

I am studying abstract algebra and I have this question: Let $G=${$a\in\mathbb{R}|-1<a<1$} Defined an operation $*$ in $G$ with $a*b=\frac{a+b}{1+ab}$ for all $a,b \in G$ Show that $(G,*)$ ...
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homotopy group of the limit space

Let $V_k(\mathbb{R}^{n+k})$ be Stiefel manifold. Using $\pi_i(V_k(\mathbb{R}^{n+k}))=0$ for all $i\leq n-1$, how to obtain $\pi_i( V_k(\mathbb{R}^{\infty}))=0$ for all $i\in \mathbb{N}$? Can I just ...
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What's the use of the reduced norm?

I've recently been introduced to the term reduced norm. I've seen it being defined in a few different ways, all of which equivalent. As it turns out, the usual norm is always the $n^{\text {th}}$ ...
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Show that $R = \{ (x,y) \in \mathbb{Z} \times \mathbb{Z} \; : \; 4 \mid(5x+3y)\}$ is an equivalence relation.

Let $R$ be a relation on $\mathbb{Z}$ defined by $$ R = \{ (x,y) \in \mathbb{Z} \times \mathbb{Z} \; : \; 4 \mid (5x+3y)\}$$ show that R is an equivalence relation. i'm having a bit of trouble ...
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Necessary and sufficient condition for a ring homomorphisms property

The question states: Let $R$ be a commutative ring with unity and let $A,B\subseteq R$ be two ideals, find a necessary and sufficient condition for $\mathrm{Hom}(R/A,R/B)=0$. Since ...
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Find a generator for $(f,g)$, two polynomials in $\mathbb Q[x]$

I have two polynomials $$ \def\f{x^5+2x^4+3x^3+3x^2+2x+1} \def\g{x^5+3x^4+4x^3+4x^2+2x+1} \def\s{\{r f + s g : r,s\in\mathbb Q[x]\}} \def\gcd{x^2+x+1} f=\f\\ g=\g $$ I want find a polynomial that ...
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Self-injective ring but not semisimple?

It is well-known that if $K$ is a field, then $K[x]/(f(x))$ is a self-injective ring for any polynomial $f(x)$ in $K[x]$. On the other hand, we know that a ring $R$ being semisimple is equivalent to ...
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Sylow subgroups problem

Let $G$ be a finite group and $p<q$ such that $p^2$ doesn't divide $|G|$. Let $H_p$ and $H_q$ be Sylow subgroups of $G$ with $H_p \lhd G$. Show $H_pH_q \lhd G \space \implies H_q \lhd G$. From the ...
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basic question of the asymptotics algebra [on hold]

I am trying to teach myself algorithms, I am following along an online web course, I need help with following questions 1) Prove that if $f(n) = O(g(n))$ and $f(n) = W(g(n))$, then $f(n) = Q(g(n))$. ...
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Cayley table for 2-bit integers ${Z_4}$

Let us consider the multiplication operation, denoted by $ \odot $ on the set of 2-bit integers ${Z_4}$ defined as follows: $$\eqalign{ & a \odot b = (ab\,\bmod \,5)\,\bmod \,4\,if\,a \ne 0,\,b ...
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Matrix in polynomial field

We are given a matrix $$M=\begin{pmatrix}0&1&1\\1&1&0\\1&1&1\end{pmatrix}$$ I need to show that $M$ represents multiplication by element $\beta $ in the field $F = ...
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Showing that $(\mathbb{R} \setminus \{ 0 \}, \, \times) \not \cong (\mathbb{C} \setminus \{ 0 \}, \, \times)$

I'm trying to show that $(\mathbb{R} \setminus \{ 0 \}, \, \times) \not \cong (\mathbb{C} \setminus \{ 0 \}, \, \times)$ as follows: note that there exists an element (namely $i$) in $\mathbb{C} ...