Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, and other advanced topics.

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Questions about functions, their domains and codomains.

I am playing around with equations about functions in general and have some questions. Question 1 If I have some functions $f,g\colon X^2 \rightarrow Y$ such that $f(x,y) = g(x,c)/g(y,c)$ then can ...
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1answer
18 views

Question about the wording of a Ring Theory problem involving ideals

The homework question is: If $I,J$ are ideals of $R$, let $IJ$ be the set of all sums of elements of the form $ij$, where $i \in I$, $j \in J$. Prove that $IJ$ is an ideal of $R$. The phrase "the ...
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The group G of isometries of R 3 which preserve a regular tetrahedron T has been shown to be isomorphic to S4. [duplicate]

The group G of isometries of R 3 which preserve a regular tetrahedron T has been shown to be isomorphic to S4. Each G-orbit of points in T is a transitive G-set. Determine all isomorphism classes of ...
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30 views

Total complex homology exact sequence

I'm been trying to do this problem (Problem 5.1.1) from Weibel's Introduction to Homological Algebra but I can't really see how to finish it. The statement of the problem is summarized as follows: ...
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1answer
21 views

Splitting a short exact sequence of orthogonal groups

How does one split the short exact sequence $$1 \rightarrow SO_n(\mathbb{R}) \rightarrow O_n(\mathbb{R}) \rightarrow \{\pm 1\} \rightarrow 1$$ ? I understand that there needs to be an injective ...
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26 views

Which one of the following ideals is radical?

For a commutative ring $A$ and an ideal $I$, $N(I)=\{x\in A\mid x^n\in I \ \mbox{for some integer}\ n\}$. Then which of these satisfy $N(I)=I$: $A=\mathbb{Z}, I=(2)$, $A=\mathbb{Z}[x], ...
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24 views

How to prove a subset is an ideal

I just started learning about Ring Theory today and I am having some trouble truly understanding and being able to apply certain concepts. The first concept I am having trouble understanding is an ...
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24 views

Question regarding character varieties on a torus with compact gauge group

Let $G$ be a compact, connected Lie group. Let $x, y \in G$ be an arbitrary pair of commuting elements. Is there necessarily a torus $T \leq G$ containing $x$ and $y$? Apparently not: Commutativity ...
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3answers
51 views

Commutator subgroup of rank-2 free group is not finitely generated.

I'm having trouble with this exercise: Let $G$ be the free group generated by $a$ and $b$. Prove that the commutator subgroup $G'$ is not finitely generated. I found a suggestion that says to ...
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25 views

Let G be a group and let x be a fixed element of G. Define Γ(x) = {g ∈ G : gx = xg} [on hold]

(a) Prove that Γ(x) is a subgroup of G. (b) Let G = A4, let x = (1 3)(2 4) and let y = (2 4 3). Find (i) Γ(x) and (ii) Γ(y).
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The Set of All Integers is NOT a Variety; How Come?

My understanding is that a variety is, essentially, a set of common "zeros" of some given functions in the given ring. My professor told us that a finite set of integers form a variety; however, the ...
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3answers
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Rigorous proof that certain ideal is not finitely generated

Let $R = F[x_1,x_2,\ldots]$ (polynomials in an infinite number of indeterminantes) and let $I = \{f \in R : f(0,0,\ldots) = 0)\}$. One can easily see that this is indeed an ideal. The proof for why ...
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1answer
33 views

The ideal $\langle x,y \rangle$ in $F[x,y]$ is not principal.

Let $F$ be a field. Apparently we know that $\langle x,y \rangle \neq \langle g(x,y) \rangle$ for any $g \in F[x,y]$. Why is this the case?
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23 views

Stochastic processes on group-valued variables

I have had this question in my head for a long time, and if I don't find out the answer I may explode. So I'm familiar with a basic Ito process, let's say: $dX_t = \mu d t + \sigma d Z_t$. There ...
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1answer
77 views

PRIMES is in P, page 4: Why is $(X+a)^{\frac{n}{p}} \equiv X^{\frac{n}{p}}+a$ implied?

PRIMES is in P, page 4, equasion (5) Edit: I should probably add that $p$ is a prime factor of some $n$. $a$ is any number from 1 to some irrelevant limit. $r$ also shouldn't matter because as far as ...
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4answers
81 views

how do I prove that $\mathbb{Q} [x]/\langle x^2 – 2 \rangle$ is a field

how do I prove that $\mathbb{Q} [x]/\langle x^2 – 2\rangle$ is a field? Is it enough to show that each since each class can be written as $[ax+b]$, then if $a=0$, $b$ is a constant which is an ...
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63 views

Help in simplifying this double summation

Can I express the following double summation $$\sum_{(i,j)\in\mathcal{R}} A_{v_i} G(v_j-v_i)$$ where $\mathcal{R}=\{ (i,j) \in \mathbb{Z}^2,i \in [1:n], j \in [1:m]\}$ while $G(.)$ is any function ...
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2answers
27 views

What is word reversal $w^R$?

In the following context, what is the formal meaning of "reversal of word $w$"? The free monoid on $A$ is the syntactic monoid of the language $\{ ww^R\ |\ w \in A^*\}$, where $w^R$ denotes the ...
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2answers
50 views

Is it possible to turn the set $\mathbb{Hom}(R,S)$ of ring homomorphisms from $R$ to $S$ into a ring?

Is it possible to turn the set $\mathbb{Hom}(R,S)$ of ring homomorphisms from $R$ to $S$ into a ring? Discuss. What I have observed that if I define the multiplication in $\mathbb{Hom}(R,S)$ s.t ...
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Prove that an equation has no elementary solution

There are methods proving that a polynomial isn't solvable in radical extensions (see Abel–Ruffini theorem) or proving that an integral or a differential equation has no solutions expressible through ...
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1answer
15 views

Multiplying Cosets

1) Let $ah$ be a coset of the subgroup $H$. Suppose there are two elements $ah_1\in aH$ and $ah_2\in aH$ such that $(ah_1)(ah_2)\in aH.$ Show that this implies that $a \in H$ and so $aH=H$. 2) ...
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1answer
30 views

permutation group

Given: Suppose G= D6, the group of all symmetries of a regular hexagon. D6={e,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b} Question: Find and list all the distinct subgroups of D6. Explain how you ...
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2answers
25 views

Parity of product of all permutations in $S_n$

Suppose $\alpha=$ the product of all permutations in $S_n$ for some $n$. For what $n$ is $\alpha \in A_n$, where $A_n$ denotes the set of all even permutations? Looking at $S_3$, I've determined that ...
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0answers
14 views

galois group of a biquadratic involving primes.

Let $f(x)=x^4-px^2+q \in \mathbb Q[x]$ be a polynomial with $p,q$ be distinct primes. Prove that $f$ it's irreducible over $\mathbb Q$. Prove that it's Galois group is the dihedral. I proved the ...
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1answer
25 views

Prove this is an automorphism

Let $r\in U(n)$. Prove that the mapping $\phi:Z_n \rightarrow Z_n$ defined by $\phi(s)=sr$ mod$n$ $\forall s \in Z$ is an automorphism of $Z_n$. My first quesiton is $U(n)$ means $U_n$ right? I have ...
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0answers
17 views

Example of a non-separable normal extension

I'm trying to give an example of a normal field extension $K|F$ that is not separable. I now that if $F$ is finite or char$(F)=0$, $K|F$ is automatically separable, thus, I must look into infinite ...
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2answers
44 views

How to build a subgroup $H\leq S_4$ having order $8$?

Can anyone explain me what would be the procedure for building a subgroup $H\leq S_4$ of order $8$? I started obviously as $H=\{id$. Then I added two disjoint $2$-cycles $(1\ 2), (3\ 4)$ for they ...
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35 views

Sufficient conditions for $G\cong N\times G/N$ [duplicate]

Given a normal subgroup $N$ of a group $G$, do there exist sufficient conditions that allow us to conclude that we have an isomorphism $$ G\cong N\times G/N?$$
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1answer
35 views

Irreducible components of an Algebraic subset.

This is question 1.27 from Fulton's textbook: http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf (the very top of page 9). 1.27. Let $V, W$ be algebraic sets in $\mathbb{A}^n(k)$, with $V\subset ...
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0answers
96 views

The group G of isometries of R3 [on hold]

I would like to request to any one in order to solve the following questions: The group G of isometries of R3 which preserve a regular tetrahedron T has been shown to be isomorphic to S4. Each ...
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1answer
26 views

Suppose two transitive G-sets X and Y are isomorphic as G-sets [duplicate]

I tried to solve the following questions. But I could not. Any one please helps me: Suppose two transitive G-sets X and Y are isomorphic as G-sets. Show that the two corresponding actions have the ...
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1answer
50 views

Operations with ideals in a commutative ring

Let $R$ be a commutative ring with identity. Let $A$ and $B$ be ideals in the ring $R$. It is true that $(A\cap B)(A+B)$ equals the product $AB$?
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14 views

Unity of a subring of $\mathbb Z_{10}$

I've been told that $S=${$[0],[2],[4],[6],[8]$} is a subring of $\mathbb Z_{10}$ with unity $[6]$. How is it true though? $[2][6]=[12]=[2]$, $[4][6]=[24]=[4]$, and so on, isn't it? I realize I'm ...
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Looking for a paper by Y. Morita

Does someone have access to the following paper? Y. Morita, Elementary proofs of the commutativity of rings satisfying $x^n=x$, Memoirs Def. Acad. Jap. XVIII (1978), 1-23. MR-Link ...
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1answer
24 views

Order of a permutation divides n in Sn

Let $\theta \in S_n$, and for any $k \in \mathbb{N}$, either $\theta^k = I_{I(n)}$ or $\theta^k$ has no fixed elements. Show that $o(\theta) | n$. $I_{I(n)}$ denotes the identity. I'm completely ...
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Textbooks to complete concurrently - Self learning empowerment

A user is completing some year challenge that takes them through $9$ textbooks and they are alternating in author. Algebra - Cohn Analysis - Rudin Topology - Lee Repeat three times. I would like ...
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1answer
56 views

Invent a linear mapping given the following conditions. [on hold]

Invent a linear mapping L such that: $L(1,2)=(3,5)$ and $L(-2,1)=(2,-3)$ I'm just unsure on how to start this problem, if anyone can give me any tips that be great thanks!
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1answer
18 views

Does the product of elements being in a group imply the individual elements are in that group?

Let $N$ and $K$ be groups and let $x\in N \cap K$ and $k\in K$. If $kx=x'k$, for some $x'\in N$, does $kx \in N \cap K$ imply that $x' \in K$?
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Let $G$ be an abelian group of order $m$. If $n$ divides $m$, prove that $G$ has a subgroup of order $n$.

So I know $|G|=p_1p_2p_3\cdots p_n$ but I don't know where to go from there. I'm having trouble figuring out how they both relate.
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three axis in $\mathbb{A}^3$ can't be defined by two functions

I am reading Shafarevich's book on Algebraic Geometry and in 1.6.5, exercise 3. He asks to prove that $X \subset \mathbb{A}^3$, which is the union of the three coordinate axis, can not be defined by ...
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2answers
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$x^2+1=0$ in $\mathbb{Z}_7$

$x^2+1=0$ in $\mathbb{Z}_7$ By trying each number, I see that there is no solution, is this correct? And could you help me with a more direct solution, since this method is not going to work for ...
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0answers
37 views

How can I determine all the subgroups of order 8 in $S_4$

Is there any way to get all subgroups of order $8$ of the symmetric group $S_4$? In general, how can I find a subgroup of specific order?
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1answer
31 views

How to show that a homomorphism between fundamental groups is an inner automorphism?

I want to show that if $\sigma$ is a loop based at a point $p \in X$, where $X$ is a topological space, then the homomorphism $\Phi _\sigma: \Pi_1(X,p) \rightarrow \Pi_1(X,p)$, is an inner ...
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29 views

Find Elements of a Quotient Group

For a group $Z_{24}$, I have two subgroups $H = \langle 4\rangle$ and $N = \langle 6\rangle$. I computed the elements of $H$ to be $\{0, 4, 8, 12, 16, 20\}$ and elements of $N$ to be $\{0, 6, 12, ...
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1answer
55 views

Prove that $S$ is an integral domain and $T$ is not an integral domain.

Let $R = \mathbb{C}[x,y]$ $R^i \subset R$ be the abelian subgroup of $R$ generated by elements of $\mathbb{C}$ times monomials of degree at least $i$ $I = (x^3+x^2-y^2)$ $S = R/I$ $S^i$ be the group ...
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1answer
33 views

Terminology of “G over H”

I am trying to find the definition of G/H (which is read as "G over H", "G modulo H", or "G mod H"). I believe that, in this case, G is a group and H is a subgroup of G.
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28 views

Suppose two transitive G-sets X and Y are isomorphic as G-sets. Show that the two corresponding actions have the same kernel. [on hold]

Suppose two transitive G-sets X and Y are isomorphic as G-sets. Show that the two corresponding actions have the same kernel. Show by example that two transitive G-sets with actions having the same ...
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0answers
20 views

Identity permutation

I was reading a proof and at one point the proof said that the identity permutation is the only permutation that fixes every element. Is the identity permutation the only permutation that fixes every ...
3
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1answer
16 views

different factors of irreducible polynomials over a Galois extension does not share roots

Let $F$ be a field and $E/F$ a galois extension. Let $f\in F[x]$ be an irreducible polynomial. I'm trying to prove that all the irreducible factors of $f(x)$ over $E[x]$ have the same degree. If ...
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0answers
27 views

Homomorphism from a subgroup to a group is injective.

I'm reading a proof and I don't quite understand one step of the proof. We want to deduce that if G acts transitively on A then $ \bigcap_{\sigma \in G} \sigma G_{a} \sigma^{-1} = 1$. (Where $G_{a}$ ...