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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$f(x)$ is still irreducible

Let $f(x) \in K[x]$ an irreducible polynomial of $K[x]$ of degree $n$. Let $K\leq F$ a field extension with $[F:K]=m$. If $(n,m)=1$ show that $f(x)$ stays irreducible also as a polynomial of ...
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How to describe the quotient group Z x Z / < (4, -6)>

While solving a problem on group theory, I encountered the quotient group Z x Z / < (4, -6)>. Here Z is the integer. At first I thought it is just Z/(4Z) x Z/(6Z). But I was wrong. the quotient ...
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Permutation Groups question in abstract algebra

Let G be a group and define a map $\lambda g : G \to G$ by $\lambda g(a) = ga$. Prove that $\lambda g$ is a permutation of $G$. Here, this is how I tried solving it. Kindly let me know if I am doing ...
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1answer
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Help with finding cosets for cyclic subgroups

The question I'm working on is: Let $G=\mathbb{Z}_4\times\mathbb{Z}_3\times\mathbb{Z}_2$ and consider the subgroup $H=\langle\left(0,1,1\right)\rangle$ of G. Find all cosets of H. So I know that in ...
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Symmetric groups and transitive action

I am trying to show that for $(x,x_1),(y,y_1)$ there exists $g\in S_n$ such that $gx=y$ and $gx_1=y_1$ where $x$, $x_1$, $y$, $y_1\in \{1,2,3,\dots ,n\}$ and $x\neq x_1$, $y\neq y_1$. Is this claim ...
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The sum of pure submodules of noncommutative ring

Let $R$ be an arbitrary noncommutative ring, and let $A$ and $B$ be pure submodules of $R$. Is the sum $A+B$ a pure submodule of $R$? I feel it is not, but I could not find out a counterexample.
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Index of intersection of subgroups in group

Let $H$ and $K$ be finite index subgroups of a group $G$ with index $h$ and $k$, respectively. I know that $H\cap K$ is of finite index in $H$ and $K$. Is the index of $H\cap K$ in $H$ bounded by ...
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1answer
34 views

Number of different magmas up to isomorphism

Let $(M,\circ)$ be a magma over a finite set of order $n$. I tried to count all the possible magmas up to isomorphism, but I just can't get it right. My naive approach was to count all the possible ...
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2answers
26 views

Symmetries of a Polynomial

I was wondering how many symmetries the polynomial $(x_1-x_2)(x_2-x_3)(x_1-x_3)$ has, and what they are. I got four: (i) $(x_1-x_2)(x_2-x_3)(x_1-x_3)$ (ii) $(x_2-x_1)(x_1-x_3)(x_2-x_3)$ (iii) ...
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30 views

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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1answer
27 views

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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1answer
34 views

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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38 views

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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2answers
14 views

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
32 views

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 Perm($\mathbb{C}$).

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 $c: \mathbb{C} \to \mathbb{C}$ ; $z \mapsto \bar{z}$. a) Let $D_n = \{ f_0, ...
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2answers
20 views

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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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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22 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
19 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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2answers
38 views

Noether normalization for $k[x]_{x}$

According to the Noether normalization theorem, there exists a $k[t]$ where $t$ is an indeterminate 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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31 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
26 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
21 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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1answer
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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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1answer
43 views

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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1answer
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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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1answer
33 views

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
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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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2answers
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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
33 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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38 views

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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0answers
15 views

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 ...
2
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1answer
24 views

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 ( ...