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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Determining the center of the p-Sylow subgroup of $S_p $

My Algebra book says without proof that the center of the p-Sylow subgroup (we will call it P) of $S_p$ is the subgroup itself. Now I can understand that P will be a subset of its center as it is of ...
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Find isomorphism for an operation

I was trying to solve this problem, but am having trouble seeing why it is an isomorphism. To map from R* to G, I think that the phi function would be Phi(x)=x/2 but that doesn't work. This phi ...
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3answers
26 views

What does it mean to evaluate a polynomial at a linear map, e.g. $p(T)$ where $p(x)$ is a polynomial?

I'm learning about the Cayley-Hamilton Theorem and the Primary Decomposition Theorem, and have trouble understanding what the notations mean. What does $\chi_T(T)$ mean? ($\chi_T(x)$ is the ...
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23 views

Direct sum notation

I was reading the direct sum of the groups and the index notation looks little bit strange for me. Group $G = \bigoplus_{\alpha < \beta}\mathbb Zx_{\alpha},$ where $\beta$ be an ordinal. Is this ...
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29 views

State a reason the given function is not a homomorphism

$f:\Bbb R \rightarrow \Bbb R$ and $f(x)=\sqrt x$ For $\forall x\lt0\in\Bbb R$, $f(x)=\sqrt x\in\Bbb C\notin\Bbb R$ Does my answer make sense, or should I elaborate with words?
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Eigenspaces of $Y$.

Given an even dimensional real vector space and a complex structure $Y$, why is its complexification the direct sum of $\text{ker}(X + iY)$ and $\text{ker}(X - iY)$?
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297 views

What does it even mean to say 'preserve structure'?

Could somebody give a concrete example of a group structure being preserved in a isomorphism, et cetera? I always hear this 'preserve structure' thing. Ok, could somebody give me a rigorous definition ...
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3answers
31 views

Showing that a subgroup of an abelian group is normal—is this sufficient?

When asked to show that a subgroup $H$ of the abelian group $G$ is normal, does it suffice to say: first, $H$ is a subgroup, so it contains the identity element of $G$ and inverses $h^\prime$ for ...
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1answer
22 views

Latin square property sufficient?

So I know that for any group table, Every row must contain distinct group elements and the same holds for every column for a group table. And this property is called the Latin Square property. ...
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21 views

A subgroup of special linear group

Does anybody know if the subgroup of diagonal and antidiagonal matrices of $SL(n,F)$ has been given a particular name? By $SL(n,F)$ I mean $n \times n$ matrices over a field $F$ with determinant 1. ...
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34 views

Equivalent condition for split exact sequence

the question is in Module Theory, Let $M,N,\&\ L$ be any R-modules. Then , for any short exact sequence $0\longrightarrow N\overset{f}{\longrightarrow} M\overset{g}\longrightarrow L ...
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1answer
27 views

Comparing two statements of Chinese Remainder Theorem (Sun-Ze Theorem)

Wikipedia states the Chinese Remainder Theorem as follows: Suppose $n_1, \dots, n_k$ are positive integers which are pairwise coprime. Then for any given sequence of integers $a_1, \dots, a_k$, ...
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92 views

Is the given subset a subspace of the given vector space?

The set of all polynomials of degree greater than 3 together with the zero polynomial in the vector space P of all polynomials with coefficients in $\Bbb R$. Let $S$ be the set of all polynomials ...
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2answers
48 views

Prove that the free group generated by two elements is a coproduct of the integers by itself in Grp

That's a problem from Algebra: Chapter $0$ by P. Aluffi, p.78, ex.5.6. One needs to prove that the group $F(\{x,y\})$ is a coproduct $\mathbb Z*\mathbb Z$ of $\mathbb Z$ by itself in the category ...
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1answer
28 views

Is every field a Krull domain?

Background: A Krull domain is an integral domain $A$ with a family $(v_i)$ of valuations on the field of fractions $K$ for $A$ satisfying the following conditions: Each $v_i$ is discrete. The ...
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1answer
34 views

Construction of free abelian group from free group

I am reading Fraleigh's Abstract Albebra recently, and I cannot prove a statement about free abelian group: Let $F[A]$ be a free group generated by set $A$ and $C$ is the commutator subgroup of ...
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48 views

Is there an example such that $\text{rank}(A^t)\neq \text{rank}(A)$?

Let $R$ be a commutative Noetherian ring and $A\in M_{n\times m}(R)$. If $R$ is a field, then $rank(A^t)=rank(A)$. However, in general Noetherian rings, does $rank(A^t)=rank(A)$ hold? Since the ...
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34 views

Maximal closed subgroups in algebraic groups

Let $G \leq GL(V)$ be an affine algebraic group, over an algebraically closed field. Say that $M$ is a proper subgroup of $G$ which is maximal among the closed proper subgroups of $G$. Does $M$ have ...
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2answers
32 views

Flat extension of noetherian rings and formal power series

Let $A \to B$ be a flat homomorphism of Noetherian rings. Is it true that it induces a flat homomorphism of formal power series $A[[x]] \to B[[x]]$?
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1answer
48 views

If $F(\alpha)=F(\beta)$, must $\alpha$ and $\beta$ have the same minimal polynomial?

Let's consider a field $F$ and $\alpha,\beta\in\overline{F}-F$ (where $\overline{F}$ is an algebraic closure). If $F(\alpha)=F(\beta)$, is it true that $\alpha$, $\beta$ have the same minimal ...
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2answers
32 views

Definition of algebraically closed field.

A field F is algebraically closed if every non constant polynomial in F[x] has a root in F. Is this the right definition? I am wondering if only one root in F and the rest of the roots not in F can ...
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25 views

Generators of the group of invertible elements of the ring $\mathbb{Z}_{14}$—are they multiplicative or additive?

When I was asked to find the generators of the group of invertible elements of the ring $\mathbb{Z}_{14}$, which are denoted as $\phi(14)$, I did not realize whether the generators are multiplicative ...
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1answer
44 views

Why is the commutator group a subgroup?

I am in Intro to Algebra, and have a question regarding the commutator subgroup. I am a bit confused with the premise, though, with how the set is a subgroup in the first place. Let $C$ be the set of ...
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1answer
17 views

Number of equivalence Relations containing $(1,2)$

Find the number of equivalence Relations on the Set $A=\{1,2,3 \}$ which contains the Element $(1,2)$. My Try: Since $(1,2)$ is to be included, so is $(2,1)$ since the Relation should be Symmetric ...
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49 views

How to compute $\mathbb Z_n \times \mathbb Z^*_m$? [on hold]

How to compute $\mathbb Z_n \times\mathbb Z_m^*$? (Here $\mathbb Z^*_m$ is the unit group mod $m$ and $(m,n)=1$.) In the paper Multiplicative properties of sets of residues it is said that by ...
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72 views

Looking for help to understand example of Group

I am looking for someone to help me to understand what is going on in the following example, from Hersteins "Topics in Algebra". It says, Let $G$ be the set of all $2*2$ matrices $$\begin{pmatrix} ...
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1answer
24 views

Unique factorization domain: $\mathbb{Z}_{n}[x]$

How to determine all $n\in\mathbb{N}$ such that $\mathbb{Z}_{n}[x]$ is a unique factorization domain? I am guessing that this would be true for all primes, since $\mathbb{Z}_n$ is a UFD when $n$ is ...
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1answer
38 views

$F[[T]] \times F[[1/T]]$, fundamental domain.

Let $p$ be a prime number. Here is a link which shows how to see that $$(\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T)))/\mathbb{F}_p[T, 1/T]$$is compact using an adelic result. (Here $\mathbb{F}_p[T, ...
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127 views

Subring of $\mathcal O(\mathbb C)$

Let $\mathfrak A \subset \mathcal O(\mathbb C)$ be the subring generated by the nowhere zero analytic functions $f: \mathbb C \to \mathbb C$. Does we have a precise description of $\mathfrak A$ ? Is ...
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$y^2 = x^3 - 26$, exist ideal satisfying conditions?

For the solution $(x, y) = (3, 1)$ of $y^2 = x^3 - 26$, does there necessarily exist an ideal $I$ of the integer ring $\mathbb{Z}[\sqrt{-26}]$ of $\mathbb{Q}(\sqrt{-26})$ such that $(y + \sqrt{-26}) = ...
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1answer
31 views

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple.

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple. I started out by trying to prove this using the Sylow theorem, but it led nowhere. I was able to ...
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1answer
57 views

Ideal in $\mathcal O(\mathbb C)$

Let $\mathfrak {I}$ the ideal generated by all the holomorphic functions which are never zero. Question : is $\mathfrak {I} = \mathcal O(\mathbb C)$ ?
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From group isomorphisms to algebra isomorphisms

Let $A$ be an algebra and let $A^{\ast}$ be the subset of units (that is, invertible elements) of $A$. Then $A^{\ast}$ is a group under the multiplication of $A$. Let $f^{\ast}:A^{\ast}\to A^{\ast}$ ...
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every $F$-algebraic homomorphism of $K$ is $1-1$ and onto? [duplicate]

Suppose that $K|F$ is a field extension of finite degree. We know that every $F$-algebraic homomorphism of $K$ is $1-1$ and onto. Also we know that every finite field extension is algebraic extension. ...
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29 views

Is it true that $M\otimes_A F\simeq M^{(I)}$?

Let $A$ be an $R$-algebra ($R$ is a commutative ring with identity $1_R$) and suppose $F$ is a left free module over $A$. Is it true that $$M\otimes_A F\simeq M^{(I)}$$ for any right module $M$ over ...
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35 views

Acting algebraically

Let $G$ be a group that acts on some non empty set $V$. What does it mean that $G$ acts algebraically on $V$? I am well aware of the definition of being algebraic. But I cant find the definition ...
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40 views

Which of the following about a permutation is correct?? (CSIR-2015, June)

Let $\sigma:\{1,2,3,4,5\}\rightarrow\{1,2,3,4,5\}$ be a permutation (one-to-one and onto function) such that $$ \sigma^{-1}(j)\le \sigma(j) \quad\text{for all $j$ such that }1\le j\le 5. $$ Then ...
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1answer
41 views

Normal Submagma?

Is there a definition of normal submagma? visit https://en.wikipedia.org/wiki/Magma_(algebra) For normal sub-quasi-group I found two: A sub-quasi-group $H$ is called normal if there exists a normal ...
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38 views

Application of the structure theorem for finitely generated modules over a PID

I've been trying to figure out the correct way to do this and I'm not sure I've got it quite right. $\mathbb{Z}^4/S$ where $S = \{(4, 12,20,8),(6,30,18,12),(4,16,16,8),(8,40,24,16) \}$ So, $M_0 = ...
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1answer
16 views

Explicitly compute the trace for the tautological representation of $D_4$ of $\mathbb{R}^2$.

Fix a finite dimensional representation $\rho: G \longrightarrow GL(V)$ of $G$. Its trace is defined as the function $tr:G \longrightarrow F$ defined by $tr(g) = tr(\rho(g))$. Explicitly compute ...
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Hardcore Abstract Algebra Book Request

I am going to start learning Abstract Algebra soon. I was originally going to start with Dummit and Foote, but I am starting to abandon that idea. I want to use a "hardcore" algebra book. I don't mind ...
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32 views

Non-zero maps between modules

Let $R$ be a ring and $M,N$ two $R$-modules such that there exists a non-zero map $\psi : M \to N$. Is it true that there exists a non-zero map $\varphi: N \to M$ ? I do not have any particular ...
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The ordinals as a free monoid-like entity

Let $\kappa$ denote a fixed but arbitrary inaccessible cardinal. Call a set $\kappa$-small iff its cardinality is strictly less than $\kappa$. By a $\kappa$-suplattice, I mean a partially ordered set ...
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Follow up question to finding primes $p$ such that $f(x)=x^6 - x^3 +1$ factors (in various ways) in $\mathbb{F}_p$

I asked this question yesterday, however, I am not sure how to compare the solution given on this site to the "worked example" solution as in my notes. $\textbf{Problem Statement:} $ Let $f(x)= x^6 - ...
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71 views

$L$-function, easiest way to see the following sum?

What is the easiest way to see that$$\sum_{(m, n) \in \mathbb{Z}^2 \setminus \{0, 0\}} (m^2 + n^2)^{-s} = 4\zeta(s)L(s, \chi)?$$Here $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to ...
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25 views

Galois group isomorphic to $\mathfrak{S}_5$.

Let $f(x) \in \mathbb{Q}[x]$ an irreducible polynomial of degree $5$ and with splitting field $K \supset \mathbb{Q}$. If $\mathbb{Q}(\sqrt{7})$ and $\mathbb{Q}(\sqrt{11})$ are subfields of $K$,is ...
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38 views

Group isomorphism exercise

I am trying to solve the following problem: Let $G$ be a finite group such that there exists an isomorphism $\phi:f:G/Z(G) \to \mathbb Z_3 \oplus \mathbb Z_3$ and $x \in G$ such that ...
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77 views

what is the greatest integer that divides $p^4-1$ for every prime number p greater than 5

what is the greatest integer that divides $p^4-1$ for every prime number p greater than 5(this is a gre subject math problem) I think that $p^4-1=(p^2+1)(p-1)(p+1)$,so 8 must divide all the $p^4-1$ ...
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1answer
37 views

Subgroup proof verification.

Let $G$ be an abelian group, K is a fixed positive integer. $H$={$a\in$ $G$ $|$ $|a|$ divides K} . Prove that $H$ is a subgroup of $G$. My way of proving (Let me know how I could make it better or ...
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1answer
56 views

Exercise from Serre's “Trees” - prove that a given group is trivial

In Serre's book "Trees" on page 10 the following exercise is given: Show that the group defined by the presentation $$x_2x_1x_2^{-1}=x_1^2, \hspace{7pt} x_3x_2x_3^{-1}=x_2^2, \hspace{7pt} ...