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Questions tagged [abstract-algebra]

Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, and modules, among other topics. Whenever you use this tag, please also include related tags like group-theory, ring-theory, modules, etc., in order to clarify which topic of abstract algebra is most related to your question and also to help other users find similar questions through the search engine.

2
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0answers
18 views

Quasi-isomorphism of an $A_\infty$ module and its cohomology

Let $A$ be an $A_\infty$-algebra over a field $k$. It is a well-known fact that $H^\bullet (A)$ also has an $A_\infty$-structure, and further one can construct a quasi-isomorphism $H^\bullet (A) \to A$...
0
votes
0answers
32 views

What can say about inverse of a generator under a homomorphism? [on hold]

Let $R$ be a commutative ring with unity and $\mathcal{I}$ be a generator of $R$. (A finite set $\mathcal I$ of ideals of $R$ is a generator if $R=\sum_{I\in \mathcal{I}}I$.) In my research, I need ...
0
votes
0answers
28 views

filtered algebra vs graded algebra

BACKGROUND When reading Okounkov-Olshanski's paper about shifted symmetric functions, they define $\Lambda^*$ to be the algebra of shifted symmetric functions. They say that $\Lambda_n^*$ is a ...
-2
votes
0answers
27 views

Vector space of polynomials degrees 0,1,2,3 [on hold]

Show that the set of polynomials of degrees 0, 1, 2 and 3 form a vector space over Q (Rational numbers).
3
votes
0answers
40 views

Does there exist some sort of classification of finite verbally simple groups?

Let’s call a group verbally simple if it does not have any non-trivial verbal subgroup. Does there exist some sort of classification of finite verbally simple groups? $G^n$, with $G$ being a finite ...
2
votes
3answers
78 views

Is an inverse element of binary operation unique? If yes then how?

I am trying to prove it but not getting any clue how to start it! $$a*b=b*a=e,$$ $$a*c=c*a=e$$ How to show $b=c$?
4
votes
1answer
40 views

Center is a normal subgroup of G

This is a problem from Herstein's Topics in Algebra. I have already shown the above result using the definition of normal subgroup. But now I want to prove it by constructing a homomorphism such ...
0
votes
2answers
47 views

Let $H$ be a subgroup of a group $G$. Show that for $a,b\in G$ we have $aH=bH$ if and only if $a^{-1}b\in H$

My Proof: Suppose $aH = bH$, then $a^{-1}aH=a^{-1}bH$ . . . $e = ab^{-1} \in H$ How to I prove conversely ($\Leftarrow$) say suppose $ab^{-1} \in H$...
0
votes
0answers
81 views

A question about dual map on modules

Let $R$ be a commutative ring (noetherian if needed), and $P$ a finitely generated projective $R$-module (of constant rank $n$). Let $P^\vee:={\rm Hom}_R(P, R)$ be the dual. For $\varphi\in{\rm Aut}_R(...
-2
votes
1answer
33 views

Proper $\Bbb Z$-submodules of $\Bbb Q$ are finitely generated or not? [on hold]

Let $M$ be a proper $\Bbb Z$-submodule of $\Bbb Q.$ Can we say that $M$ is finitely generated? I know that $\Bbb Q$ is not finitely generated as a $\Bbb Z$-module. Please help me in this regard. ...
2
votes
2answers
37 views

$\mathbb{Q}(\zeta_{10}) $ degree of field extension over $\mathbb{Q}$

I wish to calculate the degree of $\mathbb{Q}(\zeta_{10}) $ over $\mathbb{Q}$. Using the dimensions theorem: $[\mathbb{Q}(\zeta_{10}) : \mathbb{Q}]=[\mathbb{Q}(\zeta_{10}) : \mathbb{Q(\zeta_5)}] \...
3
votes
0answers
35 views

How to prove $\{G_i\to F\}$ is open covering only if $\forall$ field $K$, $F(Spec K)=\cup_iG_i(Spec(K))$?

This is an exercise in Eisenbud, Harris, Geometry of Schemes VI-11 as this part is skipped in Mumford Algebraic Geometry II. I think I figured out a way to do it but I am not totally sure. $\{G_i\to ...
3
votes
1answer
47 views

Show that if $G$ is a finite group and $H_i$ are subgroups of $G$ with $[G:H_i]=2$ then $[G:\cap H_i]=$ some power of $2$

Show that if $G$ is a finite group and $H_i$ are subgroups of $G$ with $[G:H_i]=2$ then $[G:\cap H_i]=$ some power of $2$. My try: Let the number of subgroups of $G$ be $H_1,H_2,\ldots ,H_m$ Its ...
-1
votes
3answers
23 views

Injectivity is equivalent to null space equals {0} [on hold]

Let T : V $\rightarrow$ W. Then T is injective if and only if null T = {0}. What does the null T = {0} intuitively mean here?
2
votes
1answer
33 views

Show that $Z(G) = \{ x \in G : gx = xg$ for all $g \in G$} is a subgroup of $G$. [duplicate]

Let G be a group and $g \in G$. Show that $Z(G) = \{ x \in G : gx = xg$ for all $g \in G$} is a subgroup of $G$. This subgroup is called the center of $G$. I know that associativity can be inherited ...
1
vote
0answers
25 views

Checking $A\times_B(B\times_C D)\cong A\times_C D$

Consider a category has all finite pullbacks. I want to check $A\times_B(B\times_C D)\cong A\times_C D$. It is clear from universal property of pullback that I get a unique map $A\times_B(B\times_C D)...
0
votes
0answers
13 views

Multiplicative Inverses with Equal Absolute Value

If algebraic systems may contain unequal elements of the same sign that 1) function as multiplicative inverses and 2) have the same absolute value, are there examples of such systems and ...
1
vote
2answers
51 views

If $H = \{2^k : k \in\Bbb Z\}$, show that $H$ is a subgroup of $\mathbb Q^*$.

If $H = \{2^k : k \in \mathbb{Z}\}$, show that $H$ is a subgroup of $\mathbb Q^*$. I've shown that the identity element can be found when $k=1$ because $2^1 = 2$. $1 \in H$ because $1 \in \left(\text{...
1
vote
0answers
31 views

Root space Decomposition of the A2 root system

I am looking to find the root space decomposition of the semisimple Lie algebra for the A2 root system using $L=\mathfrak{h} \oplus \bigoplus_{\alpha \in \phi } L_{\alpha}.$ ($\mathfrak{sl}(3,\mathbb{...
0
votes
0answers
27 views

Is there a program able to compute index of quotient sets and the size of the orbits?

I am interested to compute Hurwitz-Kronecker class numbers and this asks to compute a sum over representatives of a quotient set. Do you know any implemented program that does the job for any quotient ...
3
votes
1answer
72 views

$S_7$ cannot be written as a union of fewer than $459$ cyclic subgroups.

Prove that $S_7$ cannot be written as a union of fewer than $459$ cyclic subgroups. I have no idea about this. I know that $S_7$ has $7!$ elements. However, this has not been helpful. We have not ...
0
votes
0answers
15 views

Simple ring that is not semi-simple [duplicate]

Can someone give me an example of a simple ring that is not semi-simple ?
3
votes
0answers
58 views

Degree of polynomials and factors over finite fields [duplicate]

Let $p$ be a prime.Let $m,n\in \Bbb Z,m,n\ge 2$ Let $f\in \Bbb F_p[x]$ be a monic irreducible polynomial of degree $n$. Let $d$ be the number of distinct monic irreducible polynomials in the ...
1
vote
0answers
24 views

Let $n>2$ prove that $[z+\frac{1}{z}:\mathbb{Q}]=\frac{1}{2}\varphi(n)$ for every primitive $n$-th root of unity $z$

Let $n>2$ prove that $\text{deg}(m_{z+1/z}):=[z+\frac{1}{z}:\mathbb{Q}]=\frac{1}{2}\varphi(n)$ for every primitive $n$-th root of unity $z$ So, since I know that with $z$ being a primitive $n$-th ...
1
vote
1answer
51 views

Definition of sigma algebra of a group

The Wikipedia article on measurable groups says "let $(G, \circ)$ be a group and further let $\mathcal{G}$ be a $\sigma$-algebra on the set $G$." I am confused because they assume a sigma-algebra ...
-1
votes
2answers
29 views

Existence of morphism of rings [on hold]

Given any (possibly non commutative) associative ring with identity $R$, is it always possible to define a morphism of rings from $R$ to $\mathbb{Q}$ that sends $1$ to $1$?
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votes
1answer
43 views

Real numbers in theater [on hold]

I am an Italian math student; on next Wednesday, the 20th of February, I will try to qualify for the following: http://famelab-italy.it foo . Have you any suggestions on how to make interesting the ...
6
votes
1answer
63 views

Subsets of $\mathbb Z/n\mathbb Z$ disjoint with some of its shifts

Are there any descriptions of all subsets $X$ of $\mathbb Z/n\mathbb Z$ with the following property: there exists $a\ne 0$ in $\mathbb Z/n\mathbb Z$ such that $X$ is disjoint with $X + a = \{x + a \...
2
votes
1answer
52 views

Is the Quotient Group Cyclic?

I'm just wondering how to show that a quotient group $H = (G/N)$ is cyclic? Let $G= \mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$ Let $N = \left<(2,3)\right>$ , where N is a cyclic ...
2
votes
1answer
34 views

Subsets of $\mathbb Z/n\mathbb Z$ that remain disjoint with themselves under shifts

Are there any descriptions of all subsets $X$ of $\mathbb Z/n\mathbb Z$ such that for any $a\ne 0$ in $\mathbb Z/n\mathbb Z$, $X$ is disjoint with $X + a = \{x + a \pmod n\mid x \in X\}$?
3
votes
3answers
176 views

Validity of Using Induction to Show Union of an Infinite Ascending Chain of Subgroups is a Subgroup

Can this be done by induction instead of just proving the subgroup criterion? I can prove using the essentials tools of group theory, but looking at the problem, I was wondering if we can simply use ...
4
votes
0answers
77 views

Is it true, that for any two non-isomorphic finite groups $G$ and $H$ there exists such a group word $w$, that $|V_w(G)| \neq |V_w(H)|$?

Is it true, that for any two non-isomorphic finite groups $G$ and $H$ there exists such a group word $w$, that $|V_w(G)| \neq |V_w(H)|$? Here $V_w(G)$ stands for the verbal subgroup of $H$, generated ...
0
votes
1answer
62 views

Is $X^8+a \in \mathbb{F}_{49}[x]$ irreducible?

Let $f(x) = x^8+a \in \mathbb{F}_{49}[x]$ with $a \in \mathbb{F}_{49}\setminus \{0\}$. Find all $a$ such that $f$ is reducible over $\mathbb{F}_{49}$! What I know is that $\mathbb{F}_{49} \cong \...
2
votes
0answers
39 views

Order of a cyclic subgroup in G

Let G be a group where G = Z/4Z x Z/6Z. I calculated the order of G to be 24. Given the cyclic subgroup <(2,3)> in G. Can I say the order of this subgroup is also 24 given that a cyclic subgroup ...
2
votes
1answer
33 views

What is the required homomorphism satisfying $f(c)=c$ for all $c\in R$ and $f(X)=aX+b$?

My question is related to this post. I know from the Proposition that Let $φ : R → R'$ be a ring homomorphism. Given elements $a_1, · · · , a_n ∈ R'$ , there is a unique homomorphism $Φ : R[x_1, ...
1
vote
1answer
50 views

Extend order of $\mathbb{Q}$ to $\mathbb Q(X)$

I struggle to show that there exists a unique order on $\mathbb Q(X)$ that extends the order on $\mathbb Q$ such that $\forall q \in \mathbb Q, ~~~X>q$. It just means that the order of $X$ is ...
0
votes
1answer
18 views

Axioms for structures with one binary operation?

In my (limited) experience of algebra, I have only seen a handful of axioms used. The wikipedia page on outline of algebraic structures lists a table of the following axioms for structures with one ...
-1
votes
0answers
30 views

Let f ∈ Z[X] be a polynomial of positive degree. Show that there are infinitely many prime numbers p such that the polynomial f has a zero in Z/Zp. [on hold]

What is the proper way out to show for infinitely many primes.here Z/Zp is integer modulo p where p is prime.
1
vote
1answer
64 views

Why are roots of unity evenly spaced?

Roots of unity are the solutions of the complex polynomial $t^{n}-1=0$ they have the following form $E_{n}=\{e^{\frac{2\pi ik}{n}}:k\in\mathbb{Z}\}=\{e^{\frac{2\pi ik}{n}}:k=1,...,n-1\}$. From the ...
1
vote
1answer
18 views

Set of Rotations Cyclic?

For the dihedral group $D_{n}$ of order $2n$, is the group $R$ formed by its $n$ rotations cyclic in general? Or is the factor group $D_{n}/R$ cyclic? I am trying to show the series $D_{n}>R>(1)$...
-2
votes
0answers
64 views

What does the notation $G/Z(G)$ mean in group theory? [on hold]

I'm asked to prove the following: If $G/Z(G)$ is cyclic, then $G$ is abelian. But I don't know what $G/Z(G)$ means. I know that $Z(G)$ is the center of the group, explicitly given by $Z(G) = \{z ∈...
1
vote
0answers
55 views

Finding the field of fractions of a quotient of a polynomial ring?

This should be very basic but I am having a bit of trouble finding the field of fractions for quotients of polynomial rings over a field. The specific example I am having trouble with is the following:...
0
votes
0answers
30 views

Natural Transformation Between Covariant Hom-Functors

Let $\mathscr C$ be a category and $A,B\in Ob_\mathscr C$. I don't think it matters, but assume that $\mathscr C$ is locally small. I want to find a natural transformation between the covariant Hom-...
1
vote
0answers
27 views

Number of roots and degree of polynomial [duplicate]

Let $F$ be a field, and $p \in F[X]$ let be a polynomial of degree $n.$ There exists some field extension where $p$ has $n$ roots. Do you know the proof of following statement?
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vote
0answers
20 views

$R$ UFD and integral domain $\Rightarrow$ every $a\neq 0 \in R\setminus R^\times$ is a product of prime elements

Let $a$ be as in title. Then by definition of UFD we have unique factorisation $$a=\epsilon p_1\dots p_n \quad : \epsilon\in R^\times \text{ and } p_i \in R \text{ irreducible}.$$ It is known that in ...
3
votes
0answers
25 views

If $(x_n)$ is a linear recurrence, is the same true for the subsequence $(x_{pn+q})$?

Let $R$ be a ring and let $M$ be a left $R$-module. Then a linear recurrence in $M$ is a sequence $(x_n)_{n\geq 0}$ in $M$ for which there exist scalars $r_0,\ldots,r_d\in R$ such that $$x_{n+d+1} = ...
1
vote
2answers
51 views

Unique factorization and units

Wikipedia defines UFDs as A unique factorization domain is an integral domain R in which every non-zero element can be written as a product of a unit and prime elements of R. Why is there an ...
2
votes
1answer
32 views

The residue field of valuations are finite extension

Let $K\mid k$ be a finitely generated extension (maybe of transcendence degree bigger than one) and consider a rank one valuation of $K$ over $k$, that is, a function $$v:K^\times\rightarrow \mathbb{R}...
2
votes
1answer
33 views

Kummer ring - special monic polynomial with zero at root of unity

Let $ \zeta_n = e^{2 \pi i / n} $ be the n-th root of unity. Let $$ P(z) = \sum_{k = 0}^{n-1} s_k z^k $$ be a monic polynomial over $ z \in \mathbb{C} $, specified by integer coefficients $ s_k \...
5
votes
3answers
160 views

Does there exist a group that is both a free product and a direct product of nontrivial groups?

Do there exist such nontrivial groups $A$, $B$, $C$ and $D$, such that $A \times B \cong C \ast D$? I failed to construct any examples, so I decided to try to prove they do not exist by contradiction....