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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If an infinite group G has a composition series, does its subgroup necessarily have one?

If an infinite group G has a composition series, does its subgroup necessarily have a composition series? I know the answer is true for finite groups, however not sure for infinite groups. Thanks ...
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A generalization of upper nilradical

Let $R$ be a ring not necessarily commutative and not necessarily has unity. The lower nilradical of $R$ is defined by $\bigcap \text{prime ideal}$. The upper nilradical of $R$ is defined by ...
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Division of polynomials in $\mathbb{Z}[x]$

Theorem: ?For $f\in \mathbb{Z}[x]$ and $p$ prime if $\exists a\in \mathbb{Z} : f(a)\equiv 0\pmod p $ then $f(x)\equiv f_1(x)(x-a)\pmod p$ for some $f_1\in \mathbb{Z}[x]$ The proof is simple argument ...
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Universal Grobner basis

Can anyone give me a reference about the theorem " the universal Grobner basis of an ideal is finite set". Is there a proof in Sturmfels book ?
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1answer
78 views

Other than $\setminus$ and $-$, are there any other notations for the set-theoretic difference of sets?

Let $X$ denote a set, and suppose that $B$ and $A$ are subsets thereof. Then the set-theoretic difference of $B$ and $A$ may be denoted in any of the following ways: $$B \setminus A, \qquad B - A, ...
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What are the roots of the polynomial $x^{3}+3x-2\pi$ $?$

By using Dedekind's sign change rule , I can tell this polynomial $$x^{3}+3x-2\pi$$ has one real root. But I want to know what that root is and what the factorization of ...
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What is a good theoretical, yet somewhat practical, book about error correction codes?

I have started to develop some interest in error correcting codes. More Particularly I am interested in CRC's and I would like a book that treats this subject both in the theoretical aspects and in ...
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$ \Phi(G) = 1 $ or $ \Phi(G) \neq 1 $?

Let $ G $ is a finite group. Suppose $ H \unlhd G $ such that $ G/H $ is supersoluble. Suppose $ H \cap M $ is either $ H $ or a maximal subgroup of $ H $ for any maximal subgroup of $ G $. Suppose ...
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1answer
46 views

If $A$ is a commutative ring with unity, and let $a \in A$ be a nonzero element, is $\langle a \rangle$ necessarily an ideal of $A$?

My question comes from the top solution of A ring is a field iff the only ideals are $(0)$ and $(1)$. Here, at the end the solver states that $\langle a \rangle$ is automatically an ideal of $A$. Why ...
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33 views

About group actions

Let D={$n_{i}$} be a sequence of integers, $n_{i+1}$ is a multiple of $n_{i}$ ($\forall i$) and $n_{i} \to \infty$. Let us consider a group $H(D)\subset \mathbb{Z}_{n_{0}} \times ...
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Why $ K \cap H $ is a maximal subgroup of $H $?

Suppose $ G $ is a finite group and $ H \unlhd G $. Suppose that $ H \cap M $ is either $ H $ or a maximal subgroup of $ H $ for any maximal subgroup $ M $ of $ G $. Let $ N $ be a minimal normal ...
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13 views

property about centralizer of maximal subgroup

How we can show that for group $G$ (finite non-abelian p-group, I don't know which ones are necessary) and $M$ maximal subgroup of $G$. We can have $C_G(M)\le C_G(\Phi(G))\le Z(\Phi(G))$ $\Phi(G)$ ...
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What is number of group homomorphisms from $D_{12}$ to $D_{18}$?

I am willing to find out the number of group homomorphisms from $D_{12}$ to $D_{18}$ where $D_m:=\langle r_m, f_m: r_m^m=f_m^2=(r_mf_m)^2=e_m \rangle$ is the standrard dihedral group of order $2m$. ...
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19 views

Proof that $C^\infty(0,1)$ is a subring [on hold]

How do I show that the ring $C^\infty(0,1)$ of infinitely differentiable functions on the interval $(0,1)$ is a subring? Of what ring is it a subring; Map$((0,1),\mathbb{R})$? How do I show that ...
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2answers
18 views

How to prove what element in $\mathbb{Z}_n$ you get when the elements of $\mathbb{Z}_n$ are summed?

Based on trial and error I found that when $n$ is odd, the sum of the elements of $\mathbb{Z}_n$ is zero in $\mathbb{Z}_n$. When $n$ is even, the sum of the elements of $\mathbb{Z}_n$ is $n/2$ in ...
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Prerequisistes for P. May's A Concise Course in Algebraic Topology

I wonder what are the prerequisites for studying P. May's A Concise Course in Algebraic Topology. I understand basic point set topology and category theory are required. How much algebra does one need ...
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30 views

How is Buchberger algorithm a generalization of the Euclid GCD algorithm?

It is said in many places (for example, on the Wikipedia article for Buchberger's algorithm) that Buchberger's algorithm to find Groebner basis is a generalization of Euclid's GCD algorithm. This is ...
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1answer
31 views

Determinant map is homomorphism and surjective.

I just came from a course of abstract algebra, and my teacher told us that the determinant map $\det : GL(n, \mathbb{R}) \to \mathbb{R}^\times$ is a surjective homomorphism. Here, $GL(n, \mathbb{R}) ...
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2answers
41 views

Is the endomorphism ring $\mathrm{End}(\mathbb{Z}^n)$ the same as the matrix ring $\mathrm{Mat}_n(\mathbb{Z})$?

How do I show that the endomorphism ring $\mathrm{End}(\mathbb{Z}^n)$ can be identified to the matrix ring $\mathrm{Mat}_n(\mathbb{Z})$?
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42 views

Does this binary operation have a name?

Does the following binary operator on abstract linear maps $A,B:\mathbb{C}^N \rightarrow \mathbb{C}^N$, have a name: $[\{A,B\}]:= AB^{\dagger} - BA^{\dagger}$ clearly, it is real bi-linear, but not ...
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1answer
36 views

Is this still the smallest subfield of $\mathbb{C}$ that contains $\sqrt{d}$

I know from a previous question that by construction a field $\mathbb{Q(\sqrt{d})}$ is the smallest subfield of that contains $\mathbb{Q}$ and $\sqrt{d}$. But it seems to be not true if $d$ is a ...
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1answer
40 views

What are all the automorphisms of $\mathbb{Q}(\sqrt{2})$? [on hold]

The field $\mathbb{Q}(\sqrt{2})$? is defined as $\{a+b\sqrt{2}: a, b \in \mathbb{Q}\}$. Are there only two automorphisms, one mapping to $\{a+b\sqrt{2}\}$ and the other mapping to $\{a-b\sqrt{2}\}$? ...
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Subgroups of finitely generated abelian groups

Let $G\cong \mathbb{Z}_{{p_1}^{k_1}}\oplus\mathbb{Z}_{{p_2}^{k_2}}\oplus \dots\oplus\mathbb{Z}_{{p_n}^{k_n}}$. Let $H$ be a subgroup of $G$. Does $H$ necessarily have the form $H\cong ...
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3answers
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show a conclusion from the homomorphism $\phi : \Bbb R _{>0} \to \Bbb R$ such that $\phi (r) = \log(r)$

I need two show homomorphism and get a conclusion from iso1 in the following: a) I have $\phi : \Bbb R _{>0} \to \Bbb R$ $\phi (r) = \log(r)$ I assume here that $\Bbb R _{>0}$ is with ...
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19 views

Some subset is not a block in group action iff a separation property holds, questions on proof and special cases

Let $G$ be a group acting transitiviely on a set $\Omega$. A nonempty subset $\Delta$ of $\Omega$ is called a block for $G$ if for each $x \in G$ either $\Delta^x = \Delta$ or $\Delta^x \cap \Delta = ...
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23 views

$K, N < G$ and $|G:N|$ and $|K|$ coprime, then $K<N$. Group action argument?

A common exercise in finite group theory is Suppose $G$ is a finite group with $K < G$, $N \lhd G$. Suppose that $|K|$ and $|G : N|$ are coprime. Then $K < N$. The normal way to attack ...
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About $R/I$ Where $I$ is a Prime Ideal

A well known result in Commutative Algebra says: for a commutative ring $R$ with $1$, $R/I$ is an Integral Domain if and only if $I$ is a Prime Ideal of $R$. Can this result be generalised for non ...
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General way to find actions of automorphisms of the group of $x^3-2$ over $\mathbb{Q}$

I'm looking at the Galois group $\mathrm{G}(\mathbf{K}/\mathbb{Q})$ where $\mathbf{K}$ is the splitting field of $x^3-2$ over $\mathbb{Q}$. Of course, $\mathbf{K} = \mathbb{Q}(\alpha_1, i\sqrt{3})$, ...
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What is the meaning of “B is a bialgebra covariantly acting on A”?

Let $A$ be an algebra and $B$ a bialgebra. What is the meaning of "covariantly" in "B is covariantly acting on A"? Thank you very much. Edit: it is on line 13 of the abstract of the file (page 3).
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Closure of Set of Fractions with Lowest Terms Condition

Suppose I have a set of rational numbers where elements have denominators are odd and numerators and denominators are co-prime. I need to show that the set is closed under addition. It is clear that ...
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2answers
60 views

Why is Abstract Algebra so abstract? [on hold]

I mean, I haven't seen any concrete example about it. May you please give me one? Especially about Groups which we are currently having difficulties on. Thank you.
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3answers
79 views

Show that $\mathbb{Q}(\sqrt{2})$ is the smallest subfield of $\mathbb{C}$ that contains $\sqrt{2}$

This was an assertion made in our textbook but I have no idea how to show that either statement is true. Also would like to show that that $\mathbb{Q}(\sqrt{2})$ is strictly larger than $\mathbb{Q}$, ...
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Showing an iff about finitely generated modules over PID's

Let $R$ be a PID and $M$ be a finitely generated torsion $R$ module. Show that $M$ is a cyclic $R$ module iff for any prime $p \in R$ either $pM = M$ or $M/pM$ is a cyclic R module. Thoughts so far: ...
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1answer
49 views

$[G:H] < \infty$ then $gHg^{-1} = H$ and is it true that $gHg^{-1} = H$

G is a group. $H < G$ and $ g \in G$ $gHg^{-1} \subset H$ I need to prove the following : a) if $[G:H] < \infty$ then $gHg^{-1} = H$ b) without the additional fact given in (a) is it true ...
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1answer
53 views

$\mathbb{Q}\left(\sqrt{p}\right)=\{a+b\sqrt{p}\mid a, b \in\mathbb{Q}\}$ is a subfield of the field $\mathbb{R}$

Prove that $\mathbb{Q}\left(\sqrt{p}\right)=\{a+b\sqrt{p}\mid a, b \in\mathbb{Q}\}$ is a subfield of the field $\mathbb{R}$, where $p$ is a prime number I know this is true for many primes that ...
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3answers
51 views

Nilpotent ideal and ring homomorphism

In "Problems and Solutions in Mathematics", 2nd Edition, exercice 1308 Problem statement Let $I$ be a nilpotent ideal in a ring $R$, let $M$ and $N$ be $R$-modules, and let \begin{equation} f : M ...
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43 views

Finitely generated modules over principal ideal domain

Let $A$ be principal ideal domain with field of fractions $K$. $L$ is finite separable extension of $K$ and $B$ is the integral closure of $A$ in $L$. It is obvious that there exists a constant $d$ in ...
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1answer
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Find isomorphism between $D_6$ symmetries of regular triangle, $S_3$, and $GL_2(F_2)$

problem: Find isomorphism between $D_6$ symmetries of regular triangle, $S_3$, and $GL_2(F_2)$. I already proved $D_6$ is isomorphic to $S_3$. And $S_3 $ is isomorphic to $GL_2(F_2)$ Am I suppose ...
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Grothendiek and singular points

My question relates to the following thread I opened some weeks ago: A question regarding Grothendieck , topos and (adelic??) points Specifically, consider this paragraph: At 1:14:30 and after, ...
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$Y$ be the group such that it is define as $Y = < u, v | u^4 = v^3 = u= v=1, uv = v^2u^2, v^2 = v^{-1}>$ . [on hold]

Let $Y$ be the group such that it is define as $Y = < u, v | u^4 = v^3 = u= v=1, uv = v^2u^2, v^2 = v^{-1}>$ . a) Show that $v$ commutes with $u^3$. [Show that $v^2u^3v = u^3$ by writing the ...
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Supersolubility of $ G/H $ and $ H $ not deduced supersolubility of $ G $.

I want show $ S_{4} $ isn't supersoluble group. For this suppose $ 1 \leq B_{4} \leq A_{4} \leq S_{4} $ be a normal serie of $ S_{4} $, that $ B_{4} $ is Klein’s four-group. Since $ B_{4} $ isn't ...
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rings of polynomials over $Z_p$ (part-2)

An element of R is a polynomial in $x$ of degree $< r$ with coefficients from $Z_n$ (where $n$ is a composite number). We use the notation $a(x)$ to represent elements of $R$. Let $\phi :R \mapsto ...
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What is the center of $\mathbb{C}S_3$?

How do I found the center of symmetric group algebra $\mathbb{C}S_3$? and in general $\mathbb{C}S_n$? I did an example on a smaller group algebra: $\mathbb{C}S_2=\{a (1)+b(12) \mid a,b\in ...
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Term rewriting; Compute critical pairs

I have tried to solve the following exercise but I got stuck while trying to find all the critical pairs. Any help is appreciated. Let $$ \sum = \left \{ \circ, i, e \right \} $$ where $\circ$ is ...
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Prove that the order of an element in $S_n$ equals the least common multiple of the lengths of the cycles in its cycle decomposition.

Prove that the order of an element in $S_n$ equals the least common multiple of the lengths of the cycles in its cycle decomposition. proof: Let $\sigma \in S_n$. Then $\sigma = ...
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44 views

A question about injective modules [on hold]

I need to find an injective module $B$ and a submodule $A$ of $B$ such that $B/A$ not injective.
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24 views

Is this true for quotients of finitely generated abelian groups?

Let $G\cong \mathbb{Z}_{{p_1}^{k_1}}\oplus\mathbb{Z}_{{p_2}^{k_2}}\oplus\dots\oplus\mathbb{Z}_{{p_n}^{k_n}}$, where the $p_i$'s are primes. Let $H\cong ...
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1answer
26 views

Stabilization of all even/odd terms of sequence of iterated centralizers.

This is related to my previous question, see here. Fix a ring $B$. Given a subring $A \subset B$, we define$$A^! := \{b \in B : ab = ba,\text{ }\forall\,a \in A\},$$the centralizer of $A$ in $B$. ...
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39 views

Question about sub-groups. [duplicate]

Let H,K be sub-groups of G (finite order), proof that if (G;H) and (G:K) are relatively Prime G=HK. Any clue?
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31 views

Chosen maximal subject is a subgroup

Let $ G $ is a finite soluble group and $ N $ be a unique minimal normal subgroup of $ G $. Let $ G = TS $ that $ S $ is the fitting subgroup of $ G $ and $ T = N_{G}(H) $ for $ H \leq G $. Suppose $ ...