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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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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31 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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316 views

Universal Mapping Property of Free Abelian Groups

Let S be a set and $F=F_S$ the free group on S. Let $F'$ be the commutator subgroup of $F$. Set $A=A_S = F/F'$, and call it the free Abelian group on $S$. Prove the universal mapping property of the ...
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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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9 views

About $ S $-free group and normal subgroup of $ S $

Let $ S $ be a group. A group $ G $ is called $ S $-free if no quotient group of any subgroup of $ G $ is isomorphic to $ S $. Let $ G $ is finite group that is $ S $-free. if $ N \lhd S $, then is $ ...
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17 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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38 views

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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145 views

Strongly Potentially Characteristic Subgroups

Definition: A subgroup of a group is termed strongly potentially characteristic if there is an embedding of the bigger group in some group such that, in that embedding both the group and the ...
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24 views

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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5 views

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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45 views

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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105 views
+100

A question on coalgebras(2)

Assume that $C$ is a coalgebra with comultiplication $\Delta:C \to C\otimes C$. The higher order comultiplication can be defined inductively as follows(with some abuse of notations we denote them by ...
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2answers
264 views

Ring with finitely many zerodivisors

Show that a ring $ R $ with exactly $ n $ zero divisors (different from $0$) has cardinality atmost $ (n+1)^2 $. I have shown that annihilator of any element among the zero divisor is a subset ...
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1answer
39 views

Question29 from Contemporary Abstract Algebra [closed]

Consider the element A=(1101) in SL(2,R). What is the order of A? If we view A=(1101) as a member of SL(2,Zp), what is the order of A?
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3answers
90 views

For a prime integer $p$ is $pR$ a maximal ideal in $R$?

If $R$ is a commutative ring with unit and $p$ is a prime number, then is $pR$ a maximal ideal of $R$? If not what conditions should I impose on $R$?
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1answer
47 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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118 views
+100

A question on coalgebras(1)

Is there a complex coalgebra $C$ with dimension at least 2 for which the scalar operators $T(x)=\lambda x$ are the only operators which satisfy $$(T\otimes T)\circ \Delta= \Delta \circ T^{2}$$ ...
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38 views
+50

Determining the kernel of a module homomorphism

Let $p$ be a prime and let $n$ be a positive integer such that $p^n > 2$. Set $R:= \mathbb{Z}_{p^n}$, that is, the residue ring with binary operations of addition and multiplication modulo $p^n$. ...
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1answer
42 views

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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32 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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46 views

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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9 views

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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20 views

$ \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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416 views

Classify relations “is greater than or equal to”

Classify the following relations as reflexive, irreflexive, symmetric, antisymmetric or transitive. Explain each property in the context of the question. “is greater than or equal to” on the set of ...
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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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1answer
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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52 views

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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1answer
57 views

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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48 views

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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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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1answer
319 views

Proving properties of Isomorphic groups

I just wanted to practice my proofs and my understanding of Isomorphic so I decided to prove the following if I am wrong or need a better argument for anything please feel free to let me know so I can ...
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21 views

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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1answer
37 views

eigenvalues of cycle graph and its complement graph

I am trying to find the eigenvalue of cycle graph and its complement. How to simplify.Suppose $\omega^{1}+\omega^{n-1}=2\cos (2\pi/n) $, then, $\omega^{\frac{n-1}{2}}+\omega^{\frac{n+1}{2}}=\ ?$ Is ...
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2answers
74 views

What is the reduced row echelon form of $A$? [on hold]

Let $$A = \left( \begin{array}{cccc} 7 & 7 & 9 & -17\\ 6 & 6 & 1 & -2 \\ -12 & -12 & -27 & 1 \\ 7& 7 & 17 & -15\end{array} \right)$$ What is the ...
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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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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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1answer
32 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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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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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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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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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
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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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When are two direct products of groups isomorphic?

I was thinking about the following problem: Suppose that $G_1 \cong G_2$ are isomorphic groups. Under what conditions on the groups $H_1,H_2$ will we have $$G_1 \times H_1 \cong G_2 \times H_2 ?$$ ...
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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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14 views

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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$K, N < G$ 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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50 views

Deducing factorization over $\mathbb{F}_p$ from factorization over $\mathbb{Q}$

I want to show rigorously that factorization over algebraic extensions of $\mathbb{Q}$ automatically yields a corresponding factorization over $\mathbb{F}_p.$ Consider for example the polynomial ...
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2answers
40 views

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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34 views

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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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 $\mathbb{Q}$, ...