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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Additive Order and Ring

In an integral Domain, Additive order of each element is same and prime.(each equal to characteristic of ring) Is converse also true? Q1. Given additive order of each element same in a ring, would it ...
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group hyper_(Abelien_by_finite)groups has non trivial normal subgroup H of G such that H finite or Abelien

Let G be group hyper_(Abelien_by_finite)group, show that G has non_trivial normal subgroup H of G such that H finite or Abelien. hyper_(Abelien_by_finite)groups by definition if it has an ascending ...
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Ring with each element finite order but not of finite characteristic

What is example of ring without identity with each element of finite additive order but not of finite characteristic. Motivation: A ring with identity and having identity of finite additive order is ...
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How do I prove that $S_A\cong S_B\implies |A|=|B|$?

Let $A,B$ be infinite sets such that $S_A\cong S_B$. (Symmetric groups are group isomorphic) How do I prove that $|A|=|B|$? The only proof I know uses Axiom of choice. (That is, using AC to give ...
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Sign of Composition of Permutations

Let $\sigma$ be a permutation in $S_5$ with a sign of $-1$. Let $\pi$ be any other arbitrary permutation of $\{1,...,5\}$. What is the sign of the composition $\pi^{-1} \sigma \pi$? Is there any ...
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Proving the minimality of an element order

Assume that I have a finite group G of order n with a generator g, and also assume that I want to prove that $\frac{n}{gcd(n,m)} $ is the order of an element $x = g^m \in G$. First , I showed that ...
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Intuition of coset of a subgroup

Hey guys I am trying to form the intuition that distinct left coset of subgroups are actually disjoint. I understand the proof constructed but I don't think I get the intuition behind why that the ...
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Set $A$ not closed under $\star$ then $A$ not a group under $\star$?

I am currently doing some exercises. I have been through some examples of solutions in other books that questioned me. I know well that $(A,\star)$ is a group if it satisfies the following points, ...
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Invertible element of $S$

Let $S=\mathbb{Z}[\sqrt{2}]$ = {$a+b\sqrt2|a,b\in \mathbb{Z}$} and $R = \mathbb{Q}[\sqrt2]$ = {$\alpha + \beta\sqrt2 | \alpha, \beta \in \mathbb{Q}$}. Consider $x=3+2\sqrt2$ and $y = 3+4\sqrt2$ ...
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In a Category, Is the Set of Morphisms Between Objects Defined to Be All Possible Morphisms?

For instance, if I have a category $\mathfrak{M}$ whose objects are families of morphisms $\{f_i\colon A_i\to B\}_{i\in{I}}$, then if we consider two such objects, say $C=\{f_i\colon A_i\to B\}$ and ...
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Can we use the Nullstellensatz?

In $\mathbb{C}[x, y, z]$ we have that $V=\{y-x^2, z-x^3)=\{(t, t^2, t^3) | t \in \mathbb{C}\}$. To show that $$I(V(y-x^2, z-x^3))=\langle y-x^2, z-x^3\rangle $$ can we use the Nullstellensatz?? ...
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Find $f_{1}, f_{2}$ in $\mathbb{Z_{6}}[x]$ such that deg$(f_{1})$ = deg$(f_{2}) = 2$ and deg$(f_{1}+f_{2})=1$

Find $f_{1}, f_{2}$ in $\mathbb{Z_{6}}[x]$ such that deg$(f_{1})$ = deg$(f_{2}) = 2$ and deg$(f_{1}+f_{2})=1$ Find $g_{1}, g_{2}$ in $\mathbb{Z_{6}}[x]$ such that deg$(g_{1})$ = deg$(g_{2}) = 1$ and ...
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26 views

Highest Common Factor

Which is the smallest integer number $a$ so that the highest common factor of $a$ and $a+5$ is not $1$ and the highest common factor of $a$ and $a+7$ is not $1$ either? I think that it is $35$. Am I ...
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Polynomial ring addition in $\mathbb{Z_{6}}$

I know this is a very simplest question ever. But, I need help with understanding it. So here it goes... Let, $f(x) = \bar{1}+\bar{2}x+\bar{3}x^2$ and $g(x) = \bar{4}+\bar{5}x$ $\in \mathbb{Z_{6}}.$ ...
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26 views

If $G/Z(G)$ is of size $qp$ and $p-1$ is not divisible by $q$ then $G/Z(G)$ is cyclic?

I have $G/Z(G)$ with size $pq$, $p, q$ are prime and $p>q$; $(p-1) $ is not divisible by $q$ How do I deduce from the above that $G/Z(G)$ is cyclic?
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45 views

Which algebraic intuition can be used in fields

I wonder what basic laws of arithmetic of reals e.g. $x^n y^m = (xy)^{m+n}$ holds for fields. Every time I take some book on abstract algebra it proves very abstract and unpractical properties. So I ...
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26 views

Group Action and “nice” Approximation!

Studying the action of a group $G$ on a set $X$ is naturally the same as looking at the group homomorphism $\alpha: G \rightarrow Perm(X)$. So, for a given group $G$, classifying all sets $X$, on ...
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25 views

A question about $rad(M)$

We know that for any $R$-module $M$ the radical $rad(M)$ is the sum of all small submodules of $M$. My question: "if $x\in rad(M)$ is it true in general that $Rx$ is a small submodule of $M$? ...
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Irreducible polynomials of the form $x^n - q$

Is there any easy way to see the following? Let $n\in\mathbb{N}$. Let $a,b\in\mathbb{R}$ s.t. $a < b$. Then, there exists $q\in\mathbb{Q}$ s.t. $a<q<b$ and $p(x) = x^n - q$ is irreducible ...
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Find a non zero element $z\in \mathbb Z_{100}$ such that $yz = 0_{R}$ and $zy = 0_{R}$ where $y =\overline{ 14}$

Find a non zero element $z\in \mathbb Z_{100}$ such that $yz = 0_{R}$ and $zy = 0_{R}$ where $y =\overline{ 14}$ For this I have found such an element to be $\overline{50}$ since ...
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xor of three relations using Relation Algebra operations

Suppose I have three relations R1, R2, and R3. How can I specify xor of these three relations using relation algebra operations. How this scales up (for example, for four relations)? Thanks I add ...
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…and a and b are relatively prime positive integers. Find a+b. [on hold]

Let $P = \log_a b$, where $P = \log_2 3 \cdot \log_3 4 \cdot \log_4 5 \cdots \log_{2008} 2009$ and $a$ and $b$ are relatively prime positive integers. Find $a+b$.
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26 views

Why is that the radical ideal?

In my lecture notes we have the following: Definition: $f, g \in \mathbb{C}[x, y]$ $f \sim g \Leftrightarrow \exists c \in \mathbb{C}, c \neq 0$ such that $g=cf$ Example: If $f \sim g ...
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39 views

Example of a linear algebraic group which is not a Lie group

I am trying to reconcile the notions of algebraic groups, linear algebraic groups, Lie groups, and Lie algebras, along with their notions of root systems, maximal tori, etc. To begin, I am trying to ...
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82 views

What is an intuitive way to think of Cauchy's theorem?

I am looking at a problem which involves an understanding of why a finite group $G$ has an element with order $p$ if $p$ is a prime factor of $|G|$. I have looked at several resources and proofs ...
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32 views

Tensor Algebra: Symmetrization & Antisymmetrization

Problem Given the tensor algebra: $$TV:=\sum_{k=0}^\infty{\bigotimes}^k V$$ Regard the symmetrization and antisymmetrization: ...
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30 views

Find the order of $GL_2(Z_{p^{n}})$ for each prime ${p}$ and positive integer ${n}$.

Let $GL_2(Z_m)$ denote the multiplicative group of invertible $2 * 2$ matrices over the ring of integers modulo m. Find the order of $GL_2(Z_{p^{n}})$ for each prime ${p}$ and positive integer ${n}$.
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Evaluating minimal polynomial over a field $F$ as a characteristic polynomial for a $F$-linear map.

I'm considering $K/F$ to be an extension of degree $n$. I've shown that for any $a\in K$, the map $\mu_a : K → K$ defined by $\mu_a(x) = ax$ for all $x\in K$, is a linear transformation of the ...
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Proving that the irreducible polynomial of $\sqrt 2 + \sqrt 3$ over $\mathbb Q$ is irreducible modulo $p$ for every prime $p$. [duplicate]

I've computed the irreducible polynomial of $\sqrt 2 + \sqrt 3$ over $\mathbb Q$ to be $x^4+10x^2+1$. I want to show that this polynomial is irreducible module $p$ for every prime $p$. How do I do ...
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32 views

An inequality with elementary symmetric polynomials

Fix a natural number $n\geq 1$. Let $a_1, \ldots, a_n$ be $n$ real numbers such that $a_i>0$ for each $i$. Show that for each natural $k$ with $0\leq k\leq n$ $$e_k(a_1,\ldots, ...
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$V_1=V(x-y)$ and $V_2=V(x+y)$ are algebraic sets

I am looking at irreducible algebraic sets. $V \subseteq K^n$ is an algebraic set $\Leftrightarrow$ it is of the form $V(I)$, where $I=$radical Ideal of $K[x_1, x_2, \dots , x_n]$. At my lecture ...
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Algebraic set - Radical Ideal - $Rad(Rad(I))=Rad(I)$

In my lecture notes we have the following: $V \subseteq K^n$ is an algebraic set $\Leftrightarrow$ it is of the form $V(I)$, where $I=$radical Ideal of $K[x_1, x_2, \dots , x_n]$. It stands that ...
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Freely generated modules and bases

I'm trying to show that a subset $S=\{m_1,\dots,m_k\}$ of an $R$-module $M$ generates $M$ freely if and only if $S$ is a basis for $M$. I think I can see the 'if' - using the unique expression of a ...
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Determining whether ${p^{n-2} \choose k}$ is divisible by $p^{n-k -2}$ for $1 \le k < n$

Let $p$ be an odd prime and $n \ge 3$ a positive integer. I would like to know whether ${p^{n-2} \choose k}$ is divisible by $p^{n-k -2}$ for $1 \le k < n$. It should be noted that one can ...
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21 views

Algebraic Set-Radical Ideal-Nullstellensatz

In my lecture notes there is the following: $$I \rightarrow V(I) \rightarrow I(V(I))$$ It stands that in general $I \subsetneq I(V(I))$. The equality stands if and only if $I$ is a radical ...
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Isomorphism of finitely generated groups

Let G and H be two groups such that $G=<a,b>$, $H=<c,d>$ and o(a)=o(c), o(b)= o(d). Does that imply that G and H are isomorphic? or some other condition is also required ?
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23 views

A question about left ideals

Let $V$ be a finite-dimensional vector space over a field $F$. I need to prove that for every left ideal $I$ of $\operatorname{End}_F(V)$, there is only one subspace $W$ of $V$ for which $I$ = {$A ...
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Definitions of the group of cycles/group of boundaries

first I want to clarify that the class I am referring to is not really about homological algebra, rather about Galoistheory. Still we defined the group of cycles/ the group of boundaries (first ...
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Proof of the proposition $V(S)=V(\langle S \rangle )$

In my lecture notes we have the following: Proposition: $$V(S)=V(\langle S \rangle )$$ Proof: $$\langle S \rangle=\left \{\sum_{i=1}^m g_i f_i | f_i \in S, g_i \in R=K[x_1, x_2, \dots , ...
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The generator of a cyclic quotient group

(I'll use (a) to denote the subgroup of G generated by a) Let G be a finite abelian group of order n. Let a be an element of G of order k. We can easily see that (a) is a normal subgroup of G. If we ...
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Prove that each of the following sets, with the indicated operation, is an abelian group

$1.$ $x * y = x + y + k$ ($k$ a fixed constant), on the set $\mathbb R$ of the real numbers. $x * y = x + y + k = y + x + k = y * x.$ Commutativity holds. $(x * y) * z = (x + y + k) * z = (x + y + ...
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Cyclotomic polynomials and Galois groups

According to this question I want to extend the question from there. Lets consider again the galois extension $\mathbb Q(\zeta)/\mathbb Q$ where $\zeta$ is a primitive root of the $7^{th}$ cyclotomic ...
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Do these groups have a meaning?

Let $G$ be a group. We can say that $Aut(G)\leq S_G$ where $S_G$ denotes the set of all bijection from $G$ to $G$. But $S_G$ is not a good bound for $Aut(G)$ as $S_G$ grows very fast. Let's define ...
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Superfluous radical

If $M$ is a projective Artinian (left) module over a ring $R$, could one say that the radical of $M$ is a superfluous submodule in M? I know that for a projective module $M$ the radical of $M$ equals ...
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Prove $G\cong H\oplus \Bbb{Z}^{k}$.

Let $G$ be an abelian group and let $H$ be a subgroup. Let $G/H\cong \Bbb{Z}^{k}$. Prove $G\cong H\oplus \Bbb{Z}^{k}$. What I did so far is: there is an epimorphism from $G$ to $\Bbb{Z}^{k}$ such ...
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Name of this theorem? (Generalization of the class equation)

Let $X$ be a nonempty finitr set and $G$ be a finite group acting on $X$. Let $G.x_1,...,G.x_n$ be the distinct orbits of $G$. Define $F(X)=\{x\in X : \forall g\in G, g.x=x\}$. Then ...
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36 views

Ideal as kernel of a homomorphism in Gaußian integers

Consider the ring $\mathbb{Z}[i]$ of Gaußian integers. The principal ideal $(1+i)$ is maximal ideal in this ring. Since ideals are kernels of some homomorphisms, I would like to see a homomorphism ...
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24 views

Does first isomorphism theorem work both sides?

The theorem says that if I have a group homomorphism, then the kernel is normal and the image is isomorphic to the domain group modulo the kernel. Now, suppose I have $G/K \cong{H}$ where $G$ and ...
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35 views

Proving complete reducibility of modular representations

Let $G$ = $S_{3}$ and consider the $3 \times 3 $ permutation representations. For example, we have $$ \psi (123) = \begin{pmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\\ ...
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22 views

Operations with ideals: sum and product

Operations at ideals. The sum is defined as $$I_1 + I_2 + \dots + I_m =\{a_1+a_2+\cdots +a_m\mid a_i \in I_i\}.$$ It can be proven that $$I_1 + I_2 + \dots + I_m \trianglelefteq R$$ and each ...