Questions tagged [abstract-algebra]

For questions about monoids, groups, rings, modules, fields, vector spaces, algebras over fields, various types of lattices, and other such algebraic objects. Associate with related tags like [group-theory], [ring-theory], [modules], etc as necessary. To clarify which topic of abstract algebra is most related to your question and help other users when searching.

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Extension of a homomorphism

Let $G$ and $H$ be two groups such that $f:G/Z(G)\rightarrow H/Z(H)$ be an isomorphism. Now let $A$ and $B$ be normal subgroups of $G$ and $H$ respectively. If I define a map by $g:\frac{G}{A}\...
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In a Coxeter group $W$, if $w\in W$ is $I$-reduced then $\ell(vw) = \ell(v) + \ell(w)$ for all $v\in W_I$?

Let $W$ be a (finite) Coxeter group with set of simple reflexions $S$. Let $I \subset S$. We say that an element $w\in W$ is $I$-reduced if for every $s\in I$, we have $\ell(sw) = 1 + \ell(w)$. Let $...
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Natural morphism/s: The morphism doesn't preserve identity elements.

$r \mapsto (r,0)$ is the natural map 1 $r \mapsto (r,0)$ is the natural map 2 Considering the two rings $R$ and $R \times R$ the ring morphism $r \mapsto (r,0)$ is not a ring homomorphism, but it's ...
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$V(I) \neq \emptyset$ whenever $I$ is a maximal ideal of $k[x_1,\cdots,x_n]$ - Variety in a maximal - Question cannot use Nullstellensatz

I have already proved that if $V(I) \neq \emptyset$ then $V(I)$ is a unitary set. I need to conclude, using this, that any maximal ideal in $k[x_1,\cdots,x_n]$ is of the form $(x_1-a_1,\cdots,x_n - ...
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1 vote
1 answer
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when ring with roots is a field

I encountered the following problem in abstract algebra textbook. To determine if the set $$\{a+b\times 3^{1/3}+c \times 3^{2/3},\quad a,b,c \in \mathbb{Q}\}$$ is a ring, and, if so, a field as well. ...
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Upper bound on number of lattice points of given bounded convex set in $\mathbb{R}^{n}$?

Consider the bounded (closed) convex region given by: $a^Tx = b$ , $a^Tx = b + ||a||_2$ and $ m_1 \le x_i \le M_2 $ forall $1 \le i < n$ and $m_2 \le x_n \le M_2$ i.e two parallel hyperplanes at ...
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Intuition behind the elements of $\mathbb{Z}D_4$ group ring.

I have been reading Dummit and Foote, and according to it if $R$ is a commutative ring with multiplicative identity $1$, and $G$ is a finite group (say of order $n$), then the group ring $RG$ is ...
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1 answer
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Proper subfield in algebraic closed field of finite field

Let $p$ be a prime number and $\mathbb{F}_{p^n}$ denote the finite field with $p^n$ elements. Note that $\overline{\mathbb{F}_p} = \cup_{n\geq 1}\mathbb{F}_{p^n}$ is the algebraic closed field of $\...
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For what is the following theorem about Dedekind domains useful?

I am reading something about Dedekind domains and there was the following statement: Let $R$ be a Dedekind domain, then we have the following one to one correspondence $$\{\text{fractional ideals of }...
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Finite order elements of quotient group ${G}/{G^m}$

Let G be a group such that for some integer $m>1$, $(ab)^m=a^mb^m, \; \forall a,b \in G$, and let $G^m=\lbrace a^m: a\in G\rbrace$. Show that $G^m \triangleleft G$ and order of each element of ${G}/...
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2 answers
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Let $F=Q(\sqrt{2i})$,then

$F=Q(\sqrt{2i})$,then Which one of the following is not true (Duet-2017 Q.26) 1.$\sqrt{2}\in F$ 2.$i \in F$ 3.$x^8-16=0$ has a solution in $F$ 4.$dim_Q(F)=2$ I thought it as $F=${$a+b(\sqrt{2i})| a,b \...
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What are the prime ideals of $\Bbb{Z}_{(2)}[\sqrt{-3}]$

Let me consider $R=\Bbb{Z}_{(2)}[\sqrt{-3}]$, I want to find all prime ideals in $R$. I know that we have given an inclusion $$\Bbb{Z}_{(2)}\rightarrow R$$ by $$\frac{a}{b}\mapsto \frac{a}{b}\sqrt{-3}...
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Every principal ideal domain is a dedekind domain.

I thought about principal ideal domains (PID) and dedekind domains, and something confuses me a bit. I don't know if I have a thinking error. I know that every principal ideal domain is a dedekind ...
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Maximal and prime ideals in $\mathbb{Z}[x]$

I am checking if the ideals $$I=(5,x^3+2x+3), \qquad J=(4,x^2+x+1,x^2+x-1) \trianglelefteq \mathbb{Z}[x]$$ are either maximal or prime. For the ideal $I$, by factorizing $x^3+2x+3=(x+1)(x^2-x+3)$ I ...
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1 answer
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Is one-to-one-ness and operation preservation sufficient to prove an isomorphism between two finite groups with identical cardinality?

See question. In my Abstract Algebra textbook, it is mentioned that to prove an isomorphism $\phi$ exists is to show it is well defined, one-to-one, onto, and operation preserving. It seems like ...
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-2 votes
2 answers
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about finite group with order $100$.

let $G$ be a finite group with order $100$ and $H$ is a subgroup of $G$ with order $25$ also let $a \in G$ has order $50$. Now which of following options is true? 1)$\quad$ $|\langle a \rangle H |=50$ ...
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1 answer
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What does a 3-tuple mean as notation for a group?

$(F, +, 0)$ I am looking at my linear algebra lecture notes for next year and immediately came upon this notation I've never seen before, in the context of defining a field. Clearly $F$ is the set, $+$...
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1 answer
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order of an element in symmetric group .

let $f=(9~6\ 3 \ 5 \ 1 \ 4)(2\ 3 \ 4 \ 5 \ 7 \ 8 )$, $g=(4 \ 6 \ 7 \ 3 \ 1)(1 \ 4 \ 2 \ 6 \ 5 \ 7 \ 8)$ and $h=(1 \ 3 \ 5 \ 2)(2 \ 3 \ 6 \ 7)$ such that $f,g,h \in S_{9}$. find order of $g^{-2}h^{-2}...
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Class of differentiated Gamma functions: are there any algebras where they are elementary?

There is a class of differentiated gamma functions (DGFs) that basically can be expressed via derivatives of Gamma function. They include the Gamma function, Polygamma function, and Hurwitz Zeta ...
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1 vote
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How can I compute the radical $\sqrt{I}$ of an ideal $I$ of $R$?

Let $R=\Bbb{C}[X,Y]$ and $I=(XY,Y^2)$ an ideal of $R$. I want to compute $\sqrt I$. I know that $\sqrt I:=\{r\in R: \exists ~n\in \Bbb{N}, r^n\in I\}$. I clearly see that in my case $Y\in \sqrt I$ ...
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1 vote
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Correspondence theorem for ideals

Theorem. Let $A$ be an algebra and $I\subseteq A$ be an ideal. There is a one-to-one correspondence between the ideals of $A$ containing $I$ and the ideals of $A/I$. Take quotient map $\pi: A \to A/I$....
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2 votes
1 answer
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$\operatorname{tr}(A^k)=0 \space \forall k\in \Bbb{Z}^+$ implies $A$ is nilpotent. Does this imply $\operatorname {char}(K) =0$?

$\mathcal{M}_{n}(K) $: Set of all $n×n$ matrices over the field $K$. $A\in \mathcal{M}_{n}(K) $ is called nilpotent if $A^k=\textbf{0}$ for some $k\in \Bbb{Z}^+$ It is clear that if $A$ is nilpotent ...
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Check if $\Bbb{Z}[1/2]$ is local or not.

Let me consider $R=\Bbb{Z}[1/2]$. I need to check if it is local or not. I know that $R=\{p(1/2):p\in \Bbb{Z}[X]\}$ but I don't see where to start when I want to show if there is a unique maximal ...
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1 vote
1 answer
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How can I decide if $\Bbb{C}[X,Y]/(X^2,Y^2)$ is a local ring or not?

I want to check if $\Bbb{C}[X,Y]/(X^2,Y^2)$ is a local ring or not. My claim is that it is a local ring. I wanted to do it as follows: If I remember correctly we had the fact that maximal ideals of $...
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Doubt about degree for polynomimal over field?

What is the mistake of the following case:Suppose $\mathbb{F}=\mathbb{Z}_p$ and $p$ is prime.Then $x^p\equiv x$ for all $x$,then we derive $p=\text{deg}(x^p)=\text{deg}(x)=1$?
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2 votes
1 answer
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Natural partial order on idempotents and $\mathcal D$ relation

The natural partial order on the idempotents of a semigroup is defined by $$e \leq f \; \; \text {iff} \;\; ef = fe = e$$ My question regards idempotents and Green's relations: Can two different ...
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2 votes
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Linear independence of the numbers $^m\ln(a),^n\ln(a)$?

Let's assume Schanuel's conjecture is true. Let $a$ be an algebraic number, $m,n\in\mathbb{N}_+$, $^n\ln(a)=f_1(f_2(...(f_n(a))...))$ $\ \ (f_1,...,f_n=\ln)$. Can we show that the numbers $^m\ln(a),^n\...
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4 votes
0 answers
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Proof of Grün's theorem

I have a follow-up question to Problems in understanding a passage in the proof of Grün theorem for transfer. In the book of Kurzweil and Stellmacher, it is also concluded that $G\neq O^p(G) \iff H\...
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1 vote
0 answers
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Relationship involving number of inversions of permutation

Let $i,j\in \{1,\dots,n\}$ $G_{n-1}$ be the set of all $(n-1)-$permutation of $\{1,\dots,n\}\setminus\{j\}$. $C_n=\{\sigma\in S_n:\sigma(i)=j\} $ where $S_n$ is the set of all $n-$permutation of $\{...
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Existence of solutions in $\mathbb{C}(t)$ to a system of algebraic equations

In this question about finding the Galois group of the polynomial $p(x) = x^4+2utx^2+ t$ over the field $\mathbb{C}(t)$ is raised the question of whether $\sqrt t$ belongs to $\mathbb{C}(t)[\alpha]$ ...
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How to show this polynomial is primitive? [duplicate]

Suppose $f(x)=x^n+\frac{a_1}{b_1}x^{n-1} +...+\frac{a_n}{b_n}$ is a rational polynomial and $m$ is the least common multiple of the denominators $b_i$, then $mf(x)=mx^n +a_1c_1x^{n-1} +...+a_nc_n$ is ...
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Is it true that in a discrete valuation ring the maximal length of a strictly containing chain of prime ideals is one?

Let $R$ be a discrete valuation ring (DVR). I want to understand what the maximal length of a strictly increasing chain of prime ideals in a DVR is. So I want to compute the Krull dimension of $R$. I ...
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0 answers
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Prime ideals of $S^{-1}\mathbb{C}[X,Y]$ for two sets S

I want to describe the prime ideals of $S^{-1}\mathbb{C}[X,Y]$, where S is a multiplicative set, where $S:=\{X^kY^l\mid k,l \in \mathbb{N}\}$ $S:=\{fg \mid 0 \neq f\in\mathbb{C}[X], 0\neq g\in \...
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1 vote
1 answer
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Do we need the fundamental theorem of Galois theory in Galois' proof of the Abel-Ruffini theorem?

Admittedly, my question is a bit provocative. Let me be precise. The fundamental theorem of Galois theory states three things (for a fixed finite normal separable extension $M/K$): (1) The maps $L\...
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-3 votes
0 answers
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What is the order of a transposition? [closed]

Since the cycle if length2 is called transposition and it is a permutation which exchanges all elements keeping the two elements fix but i can't understand how to find the order of permutation?
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1 vote
2 answers
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Understanding properties of a matrix $A\in \mathcal{M}_n({K})$ for which $C(A)=\{f(A): f(x) \in K[x]\}$.

Consider the set of all square matrices with $n$ columns over $K(\Bbb{R} \text{ or} \Bbb{C})$: $\mathcal{M}_n({K}) $ Define $Z(\mathcal{M}_n({K})) = \{A\in \mathcal{M}_n({K}) : AB=BA, \forall B \in \...
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0 answers
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Showing that the additive group on the dyadic rationals is not a free Abelian group [duplicate]

I'm wondering whether $(\mathbb{D}, +)$ the additive group of the dyadic rationals is isomorphic to a free Abelian group or not. On an intuitive level, it seems like it's obviously nonfree, but I'm ...
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1 vote
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Find the splitting field of the polynomial $x^4-4$ over $\mathbb{Q}$ and the structure of the Galois Group.

Let, $y = x^2$. Then, $$x^4-4 = y^2 -4 = (y+2)(y-2) = (x^2+2)(x^2-2) = (x-\sqrt{2})(x+\sqrt{2})(x-i\sqrt{2})(x+i\sqrt{2}).$$ Hence, the splitting field of $x^4-4$ is $K = \mathbb{Q}(i\sqrt{2},\sqrt{2})...
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3 votes
0 answers
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Normal subgroups of direct product of non-abelian simple groups (Exercise 18 from Chapter 4.5 of Dummit's book)

I have written down a proof for an exercise from Chapter 4.5 in Dummit & Foote's Abstract Algebra. I think there might be something wrong with my proof, since most of the solutions I found ...
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5 votes
1 answer
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How do I show that the ideal $(2)$ of $\Bbb{Z}/4\Bbb{Z}$ is not flat?

Let $R=\Bbb{Z}/4\Bbb{Z}$ and let me take the ideal $(2)\subset R$. I want to show that $(2)$ is not flat in $R$. We only had the following definition: A module $M\subset R$ is flat if $M\otimes_R-$ ...
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0 votes
1 answer
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Direct product of rings: what makes the product fail to receive natural ring homomrophisms

I started out some review of math by doing abstract algebra and I do not know enough about canonical morphisms to explain this: a "ring homomorphism" $R \rightarrow R \times S$ which sends $...
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2 votes
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Property of transfer homomorphism

Let $\pi$ be a set of primes, let $P$ be a Hall $\pi$-subgroup of a finite $G$ and let $G'(\pi)$ be the inverse image of $O_{\pi'}(G/G')$ in $G$. Consider the subgroup $P^*=\langle[y,g]:y,y^g\in P, g\...
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3 votes
0 answers
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Can I construct zero-divisors from a polynomial with too many roots? [duplicate]

It is known that a polynomial in an integral domain has at most as many roots as its degree. So conversely a polynomial with more roots that its degree has to be in a ring with zero divisors. For ...
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Let $a$ and $b$ elements of order $2$ in a group $G$. Suppose $\mathrm{ord} (a,b) = 4$. Show that the subgroup generated by $a$ and $b$ is $D_4$. [duplicate]

Let $a$ and $b$ elements of order $2$ in a group $G$. Suppose $\mathrm{ord} (a,b) = 4$. Show that the subgroup generated by $a$ and $b$ is $D_4$. Problem is from Bhattacharya's book. I have a ...
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0 votes
1 answer
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Question regarding Intermediate fields and the fundamental theorem of Galois theory.

Consider the splitting field of the irreducible polynomial $x^3 - 2$ over $\mathbb{Q}$, which is $K = \mathbb{Q}(\omega,\theta)$ where $\omega$ is the cube root of $1$ and $\theta = (2)^{1/3}$. Why ...
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1 vote
0 answers
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$G'(\pi)$ is the smallest normal subgroup with an Abelian $\pi$-factor group.

I'm trying to show that following claim stated in Kurzweil and Stellmacher: Let $\pi$ be a set of primes, let $P$ be a Hall $\pi$-subgroup of $G$ and let $G'(\pi)$ be the inverse image of $O_{\pi'}(G/...
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4 votes
3 answers
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Prove $ \mbox{adj} (A) = \left[ \frac{\partial }{\partial a_{ij} } \det(A) \right]^T $

I have a bit of trouble proving an adjugate matrix equality for $A_{n\times n} = [a_{ij}]$, $$ \mbox{adj} (A) = \left[ \frac{\partial }{\partial a_{ij} } \det(A) \right]^T, \quad i, j = 1,\dots,n $$ ...
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$k$ field with characteristic $p$, polynomial $x^{p^n} - 1$ obviously has a single root, what about $x^{pn} - 1$?

Lang's Algebra (VI. 3. Roots of unity); Lang actually never mentions this as far as I know: If $k$ is a field with characteristic $p$, then the polynomial $x^{p^n} -1$ has only one root and that's 1, ...
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1 vote
1 answer
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Show that $T_{i-1}\circ d_i =d_i^\prime\circ T_i$ for $\{T_i\}$, a Chain Map from $C$ to $C'$.

$\newcommand{\Hom}{\text{Hom}}$ The following exercise is from Brown's book, A Second Course in Linear Algebra. Let $C=\{(V_i,d_i)\ |\ i\in\mathbb{Z}\}$ and $C'=\{(V_i^\prime,d_i^\prime)\ |\ i\in\...
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5 votes
2 answers
158 views

Why doesn't the condition $a/s=b/t$ iff $at=bs$ suffice when defining localization of commutative rings?

Let $A$ be a commutative ring with identity. Let $S \subseteq A$ be a multiplicatively closed set. Then the localization of $A$ by $S$ is defined as $S^{-1}A = \frac{A \times S}{\sim}$ where $(a,s) \...
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