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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Any invertible linear map is homotopic to a composition of reflections

I am trying to solve a problem in Hatcher. I reduced my problem to showing that if $f:\mathbb{R}^n\to\mathbb{R}^n$ is an invertible linear map, then $f$ is homotopic to a composition of reflections, ...
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What can you say about $f$ and $g$ in the case that $fg$ is 1) Injective, 2) Surjective - Cohn - Classic Algebra Page 15

Question: Are my proofs below valid? In both cases we are using: $f:A\to B, g: B\to C$ Notation of your type converted: $(g\circ f)(x)=g(f(x))=xfg$ If $fg$ is injective what can be said about ...
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Finding isomorphism of a factor group based on orders.

If |G|=30 and |Z(G)|=5, what commonly known group is G/Z(G) isomorphic to?
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Primary decomposition of $Z_{1001}$ as a group of multiplication

The question is asking for the primary decomposition of $Z_{1001}$ as an abelian group under multiplication. So I did the following. By Euler $\phi$ function, I count the number of integers ...
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The order of an Abelian group

Here is the question: An Abel group $G$ is generated by $x$ and $y$, with $|x| = 16, |y| = 24,x^2 = y^3$.Then what is the order of G?
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Diagrams characterizing ring characteristic, and in particular field characteristic 1?

For a given prime $p$, what is the diagram for saying (e.g. in some category where morphisms are field homomorphisms) that a field has characteristic $p$? Is there a diagram, or the shape of what ...
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Element membership for normal subgroups

Let $H$ be a normal subgroup of $G$. Let $m = (G:H)$. Show that $a^m \in H\ \forall a \in G$. I don't know how to approach this proof. I know that since $H$ is normal we have $$aH=Ha$$ $$aha^{-1}\in ...
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1answer
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How should I calculate the cosets of a subgroup of $\mathbb Z\times \mathbb Z?$

I'm trying to find the factor group $\mathbb Z^2/H,$ where $H = \{(5k,3k):k\in\mathbb Z \}.$ Would the coset of $H$ containing $(a,b)$ simply be $\{(5k + a, 3k+b):k\in \mathbb Z\}?$ If so, then how ...
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Transitive action on Poincare upper half plane

I am trying to prove that the action of $SL_2(\mathbb{R})$ on $\mathbb{H}$ via $$ \left( \begin{array}{ c c } a & b \\ c & d \end{array} \right)z\rightarrow \frac{az+b}{cz+d} $$ ...
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Show that $D_n$ is a subgroup of $\mathbb{C}$!

For $ n \in \mathbb{N}$ and $0 \le r <n$, define $f_r : \mathbb{C} \to \mathbb{C}$ ; $z \mapsto ze^{2\pi i r/n}$ and $c: \mathbb{C} \to \mathbb{C}$ ; z \mapsto z conjugate$. a) Let $D_n = \{ f_0, ...
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How to show that ideal is prime in $\mathbb{R}[x,y,z]$ modulo some other ideal

let $R:=\mathbb{R}[x,y,z]$ and $g:=x^2+y^2-z^2\in R$. I would like to know, how to show that $(x,y-z)/(g)$ is a prime ideal in R/(g), and, whether it is maximal or not. Thanks for the help!
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1answer
22 views

About primary decomposition of $\mathbb{Z}_7$

I want to find the primary decomposition of $\mathbb{Z}_7^*$ under multiplication group action. So I know 7 is a prime number so it is a group of order 6. So possibilities are $\mathbb{Z}_7^*\cong ...
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1answer
21 views

List the elements of a subgroup

The question is to list the elements of $H \subset \mathbb{Z}_{1610}$ if $H = \langle 1035\rangle$. I know that there are 14 elements in the set, I am not sure how to find the elements. Is the ...
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1answer
10 views

Polynomial rings of two variables

Prove that $(x,y)$ is not a principal ideal in $\mathbb{Q}[x,y]$. Here what is the definition of $(x,y)$? I don't know how to start the solution since I don't know the meaning of $(x,y)$.
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'The power' to inject $8$ in $5$ cannot be used to simulate the 'power' to inject $9$ in $5$

I found a cool exercise on les mathématiques (french forum); it seems to be a challenging problem with no answer for now. Let me traduce the problem. Problem. We have three set $A,B,C: ...
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2answers
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Noether normalization for $k[x]_{x}$

According to the Noether normalization theorem, there exists a $k[t]$ where $t$ is an indeterminant and $k[t]\subseteq k[x]_{x}$ is a $k$-algebra extension so that $k[x]_{x}$ is a finitely generated ...
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2answers
26 views

Isomorphism between quotient groups with normal subgroups

I'm looking at a problem in my textbook and it says: Let $ψ : G_1 → G_2$ be a surjective group homomorphism. Let $H_1$ be a normal subgroup of $G_1$ and suppose that $ψ(H_1) = H_2$. Prove or disprove ...
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Irrationality of sum of two logarithms

I try to prove that the number $$\log_2 5 +\log_3 5$$ is irrational. But I have no idea how to do it. Any hints are welcome.
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3answers
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What are all different (non-isomorphic) field structures on $\mathbb R \times \mathbb R$

We know that $\mathbb R \times \mathbb R$ forms a field under addition and multiplication defined as $(a,b)+(c,d)=(a+c,b+d)$ ; $(a,b)*(c,d)=(ac-bd,ad+bc)$ ; is there any other way to make $\mathbb R ...
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What's a semidirect product of semigroups?

I see many references to the notion in the title on the internet, but I can't find a definition. Could you please give one? A short introduction to the theory of such products (especially one relating ...
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1answer
21 views

Splitting field for polynomial

How can I find the splitting field for the polynomial $$x^{p^{50}}-1$$ ?? Could you give me some hints?? To find the splitting field for a polynomial, we have to find all the roots of this ...
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How can we solve the module function related equation?

Suppose that $\alpha,\beta=1,2,\cdots,n_1n_2$, and they satisfy the equation $$ \beta-\textbf{mod}(\beta,n_2)=\alpha-\textbf{mod}(\alpha,n_2) $$ where $\textbf{mod(,)}$ is the module function as usual ...
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Abstract Algebra and Chess [on hold]

I am currently debtating to do a Mathmatics Paper on the comparison between the game of Chess and contemporary mathematics (namely group theory). I was wondering if this has potential to be a ...
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1answer
19 views

Question on Wedderburn components of $\mathbb C[G]$

$G$ is a finite group. Wedderburn's theorem says that $\mathbb{C}G\cong R_1 \times\cdot \cdot \cdot R_r$ as rings where $R_i=M_{n_i}(D^i)$ is the ring of $n_i\times n_i$ matrices over a division ring ...
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2answers
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Quick way to determine existence of integral root of a polynomial in one variable

Suppose $p(x) \in \mathbb{Z}[x]$ and if there exist $b \in \mathbb{Z}$ s.t. $p(b)=0$, then $x-b|p(x)$. The other technique can be to put all $b \in \mathbb{Z}$. But this require to check every $b \in ...
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Check if the angle is constructable

To check if an angle $x$ is constructable do we have to use the formula $\cos{3 \theta}=4\cos^3{\theta}-3\cos{\theta}$ and find the minimum irreducible polynomial that $\cos{x}$ satisfies and if its ...
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Is an extension of a discrete absolute value discrete too?

Suppose $L/K$ is a finite extension of fields, suppose $v$ is a non-archimedean absolute value on $L$ such that the restriction of $v$ on $K$ is non-trivial and discrete. Can we say that $v$ is ...
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How do I prove $[G:H\cap K]\leq [G:H][G:K]$?

Reference: Infinite group with subgroups of finite index Let $G$ be a group. Let $H,K$ be subgroups of $G$. How do I prove that $[G:H\cap K]\leq [G:H][G:K]$? Let's not assume any index ...
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Is it true that all proper normal subgroups of $D_{24}$ abelian?

Is it true that all proper normal subgroups of $D_{24}$ abelian ? If Yes, is it true only for $D_{4n}$ groups, or for all $D_{2n}$. I was trying to list all proper normal subgroups of $D_{24}$, Using ...
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Normalizer and centralizer are equivalent when $p$ is the smallest prime dividing $|G|$

Let $p$ be the smallest prime dividing $|G|$, and suppose that some $P \in \mathsf{Syl}_p(G)$ is cyclic. Prove that $N_G(P) = C_G(P)$. So I let $G=p^\alpha m$ $p$ does not divide $m$. P is cyclic, ...
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Clarification on proof that all groups of order $< 60$ are solvable

I've manged to prove that all groups of order $< 60$ are solvable, using Burnside's theorem. However, I found an alternate proof here Question about solvable groups It states that: "Note that ...
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1answer
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Number of mutually non isomorphic Abelian groups

Let p and q be distinct primes. How many mutually non-isomorphic Abelian groups are there of order p^2q^4. I think there are 6 of them: p^2q^4 q, qp, q^2p q^2, q^2p^2 p, pq^3 pq, pq^3 q, q^3p^2 in ...
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1answer
38 views

Groups of order $2\cdot 31\cdot 61$.

What are all groups (up to isomorphism) of order $2\cdot 31\cdot 61$? Letting $n_p$ be the number of Sylow $p$-subgroups of such a group, $G$, you can show $n_{31}=1$ using the Sylow theorems ...
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Epimorphism that is not surjective in the category of Torsion Free Abelian Groups

In reading about cokernels (relating to a homework question I have) I came across the following: https://www.dpmms.cam.ac.uk/~jg352/pdf/CTSheet4-2013.pdf I specifically wondered about question 5a. ...
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$\phi: G\to G$ is a homomorphism. Denote $Ker \phi := K$ and $ Im\phi := H$. If $K \cap H =\{e\}$, can we say that $G=KH$?

While solving a problem, I came across the following question : Let $G$ be an abelian group. Suppose $\phi: G\to G$ be a homomorphism. Denote $Ker \phi := K$ and $ Im\phi := H$. If $K \cap H ...
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1answer
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Proof verification for $f$ & $g$ surjective implies $fg$ surjective - Cohn - Classic Algebra Page 15

Question: Is this a valid proof? Side question: Am I less likely to get answers based on using notation $xfg=g(f(x))$? I want to prove that if $f$ and $g$ are surjective, then $fg$ is ...
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1answer
30 views

understanding a quotient group

Let $G=\mathbb{Z}\times \mathbb{Z}$ . Let $K$ be the subgroup of $G$ generated by $(3,6)$ and $(3,1)$. Describe the rank and invariant factors of the abelian groups $K$ and $G/K$. My Try: Since $\phi ...
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1answer
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If $Q\trianglelefteq G$ is a $p$-subgroup and $P\leq G$ is a $p$-Sylow subgroup then $Q\leq P$?

Let $Q$ be a normal $p$-subgroup of a group $G$. If $P$ is a $p$-Sylow subgroup of $G$ how can I show that $Q\leq P$? Obs.: I conjecture I'll have to use the following results: (i) If $P$ is a ...
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equivalence classes partition [on hold]

Let R be an equivalence relation on A. Then show that the equivalence classes A/bar/ under this equivalence relation partitions A. Conversely, if C partitions A, define ∼ on A×A by a∼b if a,b belong ...
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Is this group cyclic?

Is this group cyclic, and what would be its generator? where H = {a+b(sqrt(2)) element of R | a, b element of Z} I know that in order for a group to be cyclic if the generator equals the group, but ...
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find variables a and b such that the identity element of the composition law $x * y = xy + xa + ay + b$ is 3

Find the variables $a$ and $b$ such that the identity element of the composition law $x * y = xy + xa +ay +b$ is $3$. I don't know how to tackle this problem, what's the reasoning for solving this ...
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1answer
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Show that it is a subfield

To show that $$\mathbb{Q}(2^{1/x}, 2^{1/y}) \subseteq \mathbb{Q}(2^{1/{xy}})$$ knowing that $(x,y)=1$, $x, y \in \mathbb{N}$ can we do the following?? $$2^{\frac{1}{x}}=2^{\frac{y}{xy}}=\left ( ...
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Find this example

Let $H=\{e,(13)\}$ be a subgroup of $S_3$. Find element $a,b \in S_3$ where $bh_2ah_1 \in aH$ but $bH\ne H$. $h_1$ and $h_2$ are elements in $H$. My friend thinks that it is (123) and (132), but I ...
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Applying Sylow Theorems to a group of order 24.

Prove that a group $G$ of order 24 with no element of order 6 is isomorphic to the symmetric group $S_4$. Here is my approach. I believe this is an alternative proof to the one provided here: Group ...
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Proving this polynomial is irreducible over any algebraically closed field.

I need to prove the polynomial $f(X,Y) = Y^2-X(X-1)(X-\lambda)$ is irreducible over any algebraically closed field $k$, for any $\lambda \in k$. I've not had any instruction in how one usually goes ...
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Division Rings and trivial ideals

I'm stuck in the following exercise, I guess this is easy but I would appreciate someone's tip.... I have to prove the following: If $R$ is a right simple ring (the unique right ideals are $R$ and ...
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Prove that there exists an isomorphism between $\mathbb Q[\alpha]$ and $\mathbb Q[\beta]$

Please, help me to understand this problem: "Let $\alpha=\sqrt[3]{2}$, $\beta=\sqrt[3]2\left( \frac{-1}{2}+\frac{\sqrt{3}i}{2} \right)$ roots of the polynomial $x^3-2$. Prove that there exists an ...
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Is an ideal also a normal subgroup?

The book I have first goes over group theory. Once it gets to rings and starts discussing subrings along with cosets and factor rings it leaves out some details for brevity and to not repeat what has ...
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1answer
20 views

Characterize semisimple rings with a unique maximal ideal

Problem Characterize the semisimple rings $R$ that contain a unique maximal ideal. I am not so sure what to do here. I know that a ring $R$ is semisimple if and only if all $R$-modules are ...
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1answer
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Prove or disprove statements about modules

I am trying to determine if the following statements are true or false (i) There are free modules with non zero elements $x$ such that $\{x\}$ is linearly dependent. (ii) There are non free modules ...