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

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The class equation of the octahedral group

I know that the class equation of the octahedral group is this: $$1 + 8 + 6 + 6 + 3$$ I think the $8$ stands for the $8$ vertices, the $6$ could be $6$ faces and $6$ pairs of edges. Then what is the ...
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0answers
29 views

A question about primitive idempotent of group algebra [closed]

How to prove: $e_j$ is primitive idempotent of group algebra $\cal{L}$ iff $\forall\ t\in \cal{L}\ $, $e_j^2=e_j$ and $e_j t e_j=\lambda_te_j$. Or in which book can I find the proof.
7
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1answer
93 views

different definitions of Hopf algebras

(i). In the book Algebraic Topology, A. Hatcher, p. 283, the notion Hopf algebra is defined as follows: (ii). However, in the book Bialgebras and Hopf algebras, J.P. May, the notion Hopf algebra is ...
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1answer
13 views

Recurrence Relations for Sequence Counting Hamming Weights

Define $a(n,k)=|\{x \in \mathbb{F}^{2n}_2 : Hx=0, ||x||=k\}|$ and $b(n,k)=|\{x \in \mathbb{F}^{2n}_2 : Hx=1, ||x||=k\}|$ where $||\cdot||$ denotes the Hamming weight of $x$ (i.e. number of non-zero ...
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0answers
54 views

Galois group over $\mathbb{Q}$ [closed]

Let $$\begin{align*} K&=\mathbb{Q}(\{\text{all $2^n$-th roots of unity for $n\in\mathbb{N}$}\})\\ L&=\mathbb{Q}(\{\text{all $n$-th roots of unity for $n\in\mathbb{N}$}\}) \end{align*}$$ What ...
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1answer
44 views

Explain why $x^4+1$ is reducible in $\mathbb{F}_p$ for any prime $p$.

We first notice that $x^4+1$ is the eighth cyclotomic polynomial, $\Phi_8(x)$. We know from the theory that $\Phi_8(\alpha)=0$ if and only if $\alpha$ has is a $8$-th primitive root modulo $p$. I've ...
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2answers
18 views

Can we embed unital Banach algebras into semi-simple ones?

A Banach algebra is (Jacobson) semi-simple if the intersection of all maximal left ideals is the zero ideal. Take a unital abelian Banach algebra $B$. Can we embed it unitally into an abelian ...
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1answer
40 views

$I\otimes I$ is torsion free for a principal ideal $I$ in domain $R$

Question is : Suppose $I$ is a principal ideal in a domain $R$. Prove that the $R$ module $I\otimes_R I$ is torsion free. Suppose we have $r(m\otimes n)=0$.. Just for simplicity assume that ...
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1answer
24 views

Intersection of subgroup and normal subgroup

I know the standard result that " let G be group and let H be subgroup and N be normal subgroup of G then H ∩ N is normal subgroup of H" but is, H ∩ N is the normal subgroup of G? if no, then is, ...
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0answers
18 views

Index of $\mathbb{Z}G$ in the maximal Order for cyclic $G$.

I am stuck at Question and I hope someone can help me. So, let $G$ be a finite cyclic group of order $n$. I am able to proof the following isomorphisms for the group algebra $\mathbb{Q}G$: ...
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18 views

Isomorphic Fields Leading to Isomorphic Splitting Fields

Link to Original Text: Theorem 10.6 Let $F, F'$ be two fields isomorphic via $\varphi$. Suppose that $f = \sum_{i=0}^m c_ix^i \in F[x]$ splits in $E$, and that the corresponding $f' = \sum_{i=0}^m ...
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0answers
39 views

Hecke algebra - independence of choice decomposition

Let $q\in \mathbb{C}$ and $\mathcal{A}_{q}$ be a Hecke algebra of degree $n$, i.e. a unital algebra generated by elements $\sigma_1,...,\sigma_{n-1}$ satisfying the following relations ...
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1answer
31 views

Let N = Fit(G). Why $ N = O_{p}(G) $ and $ A \leq Z(N) $?

Let $ G $ be a soluble group and $ A $ be a minimal normal subgroup of $ G $,where $ A $ is an elementary abelian group of prime power order. Let each chief factor of $ G/A $ has order $ 4 $ or a ...
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0answers
36 views

The group $G$ has order 56. Show that it contains normal Sylow p-subgroup.

I'm new at this topic, so I'm not sure about whether is my solution acceptable or not. Could you check it? By Sylow theorem there exist $|P|=7$ and $|Q|=8$. Sylow $p$-subgroup is normal iff it is ...
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0answers
22 views

How to calculate the $k$-dimension of a subspace of a polynomial ring?

Let $k$ be an infinite field and $R:=k[x_1,...,x_n]$ the polynomial ring in $n$ indeterminates. Why is the $k$-dimension of $U$ given by $\begin{pmatrix} n+m-1 \\ m\end{pmatrix}$, when $U$ is the ...
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4answers
47 views

Prove that the 4-group V is normal subgroup of $S_4$ by using isomorphism theorem

Prove that the 4-group V is normal subgroup of $S_4$ First, by using the multiplication table, I am able to prove that 4-group V is subgroup of $S_4$. But I face problem in proving that $\forall ...
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2answers
45 views

Prove that if $G$ has a subgroup of order $n$, then $H$ has a subgroup of order $n$.

Would someone please help me: Question : Let $φ : G → H$ be an isomorphism of two groups. Then prove that if $G$ has a subgroup of order $n$, then $H$ has a subgroup of order $n$. Proof : For the ...
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1answer
22 views

Extension of field automorphism to automorphism of algebraic closure

Let $k$ be a field and let $f(x)\in k[x]$ be irreducible. Let $K$ be the algebraic closure of $k$, and say among the roots of $f(x)$ are $\alpha,\beta\in K$. Then there exists an automorphism of $K$ ...
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3answers
32 views

Finding Linear Combination of Polynomials

I am stuck on a question involving finding the greatest common divisor of polynomials and then solving to find the linear combination of them yielding the greatest common divisor. My work thus far is ...
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2answers
70 views

Every isomorphism of subgroups of $\mathbb{Q}$ is of the form $\varphi(x)=qx$ for some $q \in \mathbb{Q}$.

Let $A$ and $B$ be subgroups of the additive group $\mathbb{Q}$. If $A$ is isomorphic to $B$ and $\varphi : A \rightarrow B$ an isomorphism, then show there is $q \in \mathbb{Q}$ such that ...
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1answer
24 views

Projection of a discrete subgroup of $R^n$ [duplicate]

Let $A$ be a discrete subgroup of $\Bbb R^n$ and let $V$ be a $m<n$ dimensional $\Bbb R$-subspace of $\Bbb R^n$. Is the projection of $A$ onto $V$ a discrete subgroup? I am most interested in the ...
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1answer
16 views

Semidirect product defined by a non-trivial abstract homomorphism

Let's say we are given two abelian subgroups, $H$ and $K$ of a group $G$. It is obvious that the semi-direct product is abelian (i.e. the direct product) if the homomorphism $\phi: K \to Aut(H)$ is ...
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2answers
63 views

If $X^p-a$ has no zeros in a field $F$ of characteristic $p$ where $a \in F$, is it irreducible?

Let $F$ be a field of characteristic $p>0$ and $a\in F$. I have an easy question which I'm stuck on. If the polynomial $X^p-a$ has no zeros in $F$ then is it irreducible over $F$? ...
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1answer
40 views

Characterizing Subgroups through Homomorphisms

Just for fun, I was wondering how one could specify subgroups using homomorphism equations. For instance, for vector spaces, every subspace can be specified as the kernel of some homomorphism. For ...
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56 views

When does a Galois group of a quintic have order divisible by three?

Apparently, nice necessary and sufficient conditions are known for a Galois group of a degree 5 polynomial to have order divisible by 3. What are these conditions? The possible Galois groups for an ...
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2answers
21 views

Quotient of space and a group of maps, Riemann surfaces

I've been attempting to study Riemann surfaces, and I have continuously run into this notion which eludes me. I see people write things like $ \mathbb H / <z\mapsto z+1>$ or $\mathbb D / PSL$. I ...
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2answers
72 views

Elements of order $3$ in $\text{Aut}\left(\mathbb{Z}/91\mathbb{Z}\right)$

It looks like someone has already been here, but the question I have goes farther. To summarize my work, as well as the work in the above post, we know that ...
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0answers
25 views

Given $r \in \mathbb{N}$, prove there exist odd primes $p,q$ s.t. $p$ splits into $r$ primes in the $q$th cyclotomic field.

Given $r \in \mathbb{N}$, prove there exist odd primes $p,q$ s.t. $p$ splits into $r$ primes in the $q$th cyclotomic field. Let $\omega = e^{2\pi i /q}$. Then I know that $(p,q) = 1 \implies p ...
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49 views

The Fixed Point Theorem in Artin's book

Theorem 7.3.2 Let G be a p-group, and let S be a finite set on which G operates. If the order of S is not divisible by p, there is a fixed point for the operation of G on S - an element s whose ...
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0answers
87 views

On a theorem of Akizuki concerning the minimal number of generators of an ideal

I am looking for a theorem of Akizuki I was told by my professor. He said me that Akizuki showed in his paper "Zur Idealtheorie der einartigen Ringbereiche mit dem Teilerkettensatz" (1938) a result ...
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0answers
33 views

Let $ G $ is finite nilpotent group. can we say $ G $ is $ p $-supersoluble for any prime $ p $?

Let $ G $ is finite nilpotent group. Thus $ G $ is $ p $-nilpotent group for any prime $ p $. Now can we say $ G $ is $ p $-supersoluble for any prime $ p $?
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3answers
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Why do we show that structures aren't isomorphic by exhibiting a property not shared by one of them?

If someone asks me how to prove that two order structures $\langle A,\leq \rangle$ and $\langle B,\preceq \rangle$ are isomorphic I would immediately suggest: try to find a function $f:A\to B$ such ...
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2answers
35 views

How to prove generator is is normal subgroup of G iff a is the center of G?

Problem: Let $G$ denote a group, and $H$ a subgroup of $G$. Suppose $a$ to be an element of $ G$ of order 2. Prove that $\langle a \rangle$ is a normal subgroup of $G$ iff a is in the center ...
3
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2answers
37 views

if $H \leq G$ has index 2, then $a^2\in H$ for every $a\in G$

if $H \leq G$ has index 2, then $a^2\in H$ for every $a\in G$ I am not sure that whether the way that i prove this statement is correct. Since $[G:H]=2, \forall a\in G,G/H=\{H,Ha\}$ Hence $Ha^2=H ...
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1answer
45 views

Priority of the 3 axioms of groups [duplicate]

In my book about Abstract Algebra, it is stated that A group $\langle G,*\rangle$ is a set $G$, closed under a binary operation $*$, such that these 3 axioms are satisfied: $g_1$: For all ...
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3answers
177 views

Can $1=0$ ever make sense?

Can $1=0$ ever make sense? In more detail, under which interpretation of $0$ and $1$ is $1=0$ possible? what are the consequences of such a result? Under which interpretations would $1=0$ never be ...
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1answer
45 views

Annihilator of maximal ideals in a finite dimensional algebra

I wonder if the following is correct: The left (right) annihilator of every (2 sided) maximal ideal in a finite dimensional $k$-algebra is always nonzero. Clearly this is true for semi-simple ...
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3answers
79 views

When is a given pair $(G,*)$ a group?

I'm doing practice problems for my linear algebra class, and I don't understand how to use the group axioms to see which pairs $(G,*)$ are groups. $G= (0,\infty)$ with $*$ given by addition $G= ...
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22 views

Classification of commutative ring ideal closure operators?

First, some setup: So: given a commutative ring $R$, let $Ideals(R)$ be set of ideals of $R$ and let $IdealClosure(R)$ be the set of closure operations $cl: \mathcal{P}(R) \rightarrow Ideals(R)$. In ...
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1answer
26 views

Simple bimodule over a basic algebra

I am looking for a reference for the following result: Let $k$ be an algebraically closed field and $A$ be a finite dimensional $k$-algebra. If $A$ is basic, then every simple $A$-$A$ bimodule is ...
4
votes
1answer
76 views

Mason's Theorem Proof. From Algebra (S. Lang)

Mason's Theorem Proof. I have a question about the last step in the proof. I will write it. Mason's Theorem states that if $a, b$ and $c$ are relatively prime polynomials such that $a + b = c$ ...
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0answers
39 views

Problem involving cubic field extensions

Let $F$ be a field of characteristic $0$ and let $L$ be a cubic extension. I want to show that there exists an element $a \in F,$ and an extension $L_0$ of $\mathbb{Q}(a)$ such that ...
7
votes
1answer
143 views

Reading mathematics at the graduate level - does every single proof matter?

I would be interested in hearing from PhD students (and upwards) specializing in pure maths (in particular, the more algebraic aspects). My question is this: When reading to learn mathematics at ...
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0answers
33 views

Universal Algebra: partial algebras and homomorphisms

I'm looking for an accessible introductory text on universal algebra, one which discusses homomorphisms in the context of partial algebras. I have been recommended the Grätzer book, "Universal ...
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1answer
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52 views

Ring Theory: Identity Elements

A brief question$:$ If a ring is specified to have an identity, is it implicit that the identity in question is the multiplicative identity? From the definition of a ring $R$, $R$ must contain ...
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1answer
83 views

Is the cuspidal curve $\mathcal{M}$ is a coarse moduli space for lines in $\mathbb{C}^2$?

As the question suggests, is the cuspidal curve $\mathcal{M}$ a coarse moduli space for lines in $\mathbb{C}^2$? I'm inclined to believe the answer is no, but all attempts at proving it so far have ...
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1answer
18 views

Show that in ascending Loewy series, $S^r(R)=R$

Let $R$ be an Artinian ring, $N$ its radical, and $r$ the smallest natural number such that $N^r=0$. Define an ideal $S^n(R)$ of $R$ recursively as follows: $S^1(R)=soc(R)$ Assuming ...
3
votes
1answer
42 views

Tensor product isomorphic to a free module

Is it true that if $R$ is a domain with quotient field $K$ and $M$ is a finitely generated torsion-free $R$-module then $M\otimes_R K$ is isomorphic with $K^n$ for some $n$? I know that the first is ...
4
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2answers
358 views

Proof of Wilson's Theorem using concept of group.

I am studying group theory so I do it by using the concept of group. What I am trying to prove is if p is prime then $(p-1)!\equiv-1\mod p$ Note that $\mathbb{Z_p}$ forms a multiplicative group. ...