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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5
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1answer
37 views

The centralizer of $x_3x_2x_1x_2x_1x_3^{-1}$ in $F_3$ is $\langle x_3x_2x_1x_2x_1x_3^{-1}\rangle$?

The centralizer of $x_3x_2x_1x_2x_1x_3^{-1}$ in $F_3$ is $\langle x_3x_2x_1x_2x_1x_3^{-1}\rangle$. The centralizer of an element in $F_3$ is the set of elements of $F_3$ that commute with that ...
5
votes
4answers
52 views

Let $H$ and $K$ be subgroups of $G$ such that $H \cap K = \{e\}$. Then $H \cup K$ is a subgroup of $G$?

Let $H$ and $K$ be subgroups of $G$ such that $H \cap K = \{e\}$. Then $H \cup K$ is a subgroup of $G$. I know that $H \cup K$ is a subgroup of $G$ if and only if $H \subseteq K$ or $K ...
7
votes
0answers
56 views

A List of Standard or “Cliche” Homeomorphisms [duplicate]

Learning topology has been hard. I just cannot see how some people can come up with complex functions that link one space to another, in a homeomorphic sense. The explanations are always "Well if you ...
3
votes
0answers
31 views

Computing center of an algebra

Let us define an associative algebra over $\mathbb{C}$ with generators $x, y, z$ and the following relations: $x^2=x, y^2=y, z^2=z, 2yxy=y, 3zyz=z, xz=zx$. I am interested in finding center of this ...
1
vote
2answers
343 views

Does there exist a p-group of order 99?

Does there exist a p-group of order $99$? We can first observe that $99=3^2 \times 11$. I then believe we need to apply Sylow's theorem but I am not sure how, exactly. How can I prove existence ...
5
votes
1answer
57 views

Show that the number is irrational $\forall n$

I need to show that the number $\sqrt 2+ \sqrt[3]{3}+\sqrt[4]{4}+\sqrt[5]{5}+...+\sqrt[n]{n}$ is irrational for any n, and I don't have a clue about how I could show that. Thank you!
3
votes
2answers
36 views

Order of $G$ is 35, are there elements of order 5 and 7 in G?

I have a group of order 35 and I want to know if it contains elements of order 7 and 5. I know that it does and there is a proof that is much longer, but I wanted to know if the following worked to ...
1
vote
2answers
27 views

Why is $\varphi(X_i) = X_i + b_i$ an automorphism of $K[X_1,\dots,X_n]$?

I'm trying to justify to myself the assertion (used here) that given a field $K$ and elements $b_1,\dots,b_n\in K$, the map $\varphi(X_i) = X_i + b_i$ is a $K$-automorphism of $K[X_1,\dots,X_n]$. ...
-2
votes
1answer
26 views

Fide the degree of the splitting field of a polynomial over $Q$

I want to know how to determine the degree of the extension $K/\mathbb{Q}$, where $K$ is the splitting field of the polynomial $x^6+1\in \mathbb{Q}[x]$ over $\mathbb{Q}$.Do I have to get all the roots ...
6
votes
1answer
76 views

Is it possible for $R \oplus M$ and $R \oplus N$ to be isomorphic to each other if $M$ and $N$ are not isomorphic?

Suppose $M$ and $N$ are non-isomorphic $R$-modules (where $R$ is a commutative ring with a unit element).Can we conclude that $R \oplus M \not\simeq R \oplus N$ ? If not in this most general ...
1
vote
0answers
13 views

trying to understand the Jacobi identity for vertex operator algebras

I was having a look at James Lepowsky and Haisheng Li's "Introduction to Vertex Operator Algebras and their Representations" and when they give the Jacobi identity they have ...
3
votes
1answer
48 views

Rings in which $ab=0$ implies $axb=0$

I'm sure there must be some standard term for (not necessarily commutative) rings $R$ in which $ab=0$ implies $(\forall x)\, axb=0$ (for example, this is the case if $R$ is commutative or is a ...
-2
votes
1answer
43 views

Is $G/N$ isomorphic to $\mathbb R ?$

$$G=\left\{\begin{bmatrix} a & b \\ 0 & \ \ \ \ a^{-1}\end{bmatrix} : a,b \in \mathbb R;a>0 \right\}$$ $$ \hspace{-1.1in}N=\left\{\begin{bmatrix} 1 & b \\ 0 & 1 \end{bmatrix} :b\in ...
1
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0answers
24 views

A better way of showing the set of all $m\times n$ matrix is an $R$ module.

Let $R$ be a ring and $m$ and $n$ be any positive integers. For any $a\in R$ and $A=(a_{ij})\in M_{m\times n}$ define $aA=(aa_{ij})$. Then prove that $M_{m\times n}$ is an $R$-module. Edited: Take ...
0
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2answers
28 views

$\mathbb F_q[x]/(p(x))$ is a field of order $q^n$.

Let $\mathbb F_q$ be a field of order $q$ and $p(x)$ be an irreducible element in $\mathbb F_q$ of degree $n$. Then prove that $\mathbb F_q[x]/(p(x))$ is a field of order $q^n$. Attempt: As $p(x)$ ...
4
votes
0answers
22 views

Determine the center of the group $GL_n(\mathbb{R})$? [duplicate]

Determine the center of the group $GL_n(\mathbb{R})$. The center of a group $G$ is the set of elements that commute with every element of $G$. I think the answer is $Z(GL_n(\mathbb{R}))=\{\lambda ...
-2
votes
1answer
34 views

Prove that: $0\rightarrow B' \overset{i}{\rightarrow} B \overset{p}{\rightarrow} B'' \rightarrow 0$ is a split short exact sequence [on hold]

Suppose map $i^{*}$ such that $Hom_{\mathbb{R}}(B,M) \overset{i^{*}}\rightarrow Hom_{\mathbb{R}}(B',M)$ is surjective for every $M$. Prove that: $0\rightarrow B' \overset{i}{\rightarrow} B ...
2
votes
1answer
34 views

Every module free implies no nonzero maximal ideal.

Show that given a ring $R$ with identity such that every $R$-module is free, then $R$ has no nonzero maximal ideals. I only know that every ideal of $R$ is a direct summand of $R$. Is it ...
0
votes
0answers
33 views

Find the lattice of Galois Field

I am wondering what the lattice of subfield of $GF(p^{30})$ looks like. I know that it starts from $GF(p)$ and then $GF(p^2)$ and $GF(p^3)$, but then I am lost. And I looked it up online, but can't ...
0
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0answers
13 views

degree of a field extension and generator

I assume that $\alpha$ is algebraic over F and then $F(\alpha)$ is just a simple extension. And then I am able to find a basis for both $F(\alpha)$ and $F(\alpha^3)$. Then we can apply the theorem ...
3
votes
1answer
43 views

Rate of a binary linear code given all code words have weight integer multiple of 4

Supposed C is a linear binary code with the property that each code word is of Hamming weight n*4 (that is every word has a Hamming weight that is an integer multiple of 4). Show that the rate of C is ...
6
votes
3answers
279 views

Prove that any two cyclic groups of the same order are isomorphic?

Prove that any two cyclic groups of the same order are isomorphic. Let the groups be $G,H$ with order $k$. Let $G=<a>$ and $H=<b>$. Thus we have $|a|=|b|=k$ and by definition, ...
0
votes
2answers
34 views

Degree of extension $\mathbb{Q}(\sqrt{2},\sqrt{6})$ over $\mathbb{Q}(\sqrt{3})$

Find degree of extension$[\mathbb{Q}(\sqrt{2},\sqrt{6}):\mathbb{Q}(\sqrt{3})]$. I think that $\sqrt{6}=\sqrt{2}\sqrt{3}$, then a basis of vector space $\mathbb{Q}(\sqrt{2},\sqrt{6})$ over field ...
0
votes
0answers
27 views

Primordial elements of a vector space

We were given the following problem in our Algebra class. Let $V$ be a $K$-vector space (not necessarily finite dimensional), and fix a basis $(e_i)_{i \in I}$ of $V$. If $x = \sum \xi_ie_i \in V $, ...
2
votes
0answers
28 views
+50

Matrix representations of the generators of the full octahedral group

I want to find matrix representations of the generators of the full octahedral group which has the presentation $\{a,b,c|a^2=1,b^3=1,(ab)^4=1,ac=ca,bc=cb\} $ where a,b and c are the generators of the ...
1
vote
1answer
16 views

Why doesn't the coset (1,4,2,3)K belong to the Quotient group

I had been given the following question and answer ( in the image) However i do not understand, why for example: (1,4,2,3)K does not belong the the quotient group? Is there any faster way of ...
4
votes
2answers
59 views

What's the order of a semigroup?

For a group the order, of an element is the smallest positive integer m such that a^m = e. But what's the order of an element of a semigroup? Or there isn't anything like that?
0
votes
1answer
26 views

Show that $\operatorname{End}_A(P)$ is a central, simple $K$-algebra if $P$ is a f.g. projective $A$-module

Let $A$ be a central, simple $K$-algebra and let $P$ be a finitely generated projective $A$-module. I want to show that the endomorphism ring $\operatorname{End}_A(P)$ is also a central, simple ...
3
votes
1answer
40 views

Show that any group of order 20 is not simple?

Show that any group of order $20$ is not simple. Denote the group $G$. It seems intuitive to state first that $20=2^2 \times 5$. Sylow's theorem then states that since a prime number, $5$, ...
1
vote
1answer
31 views

Direct Product of Completions

This question is regarding Theorem 8.15, page 62 of Matsumura's Commutative Ring Theory. It says that if $A$ is a semi-local ring and $I=m_1\cdots m_r$ be the Jacobson radical of $A$. Then ...
-1
votes
1answer
44 views

Finding subgroup [on hold]

Find a subgroup $\mathrm{H}$ in group $\mathbb{Z^2}$ so that $\mathbb{Z^2}/\mathrm{H} \simeq \mathbb{Z_6} \times \mathbb{Z_{10}} \times \mathbb{Z_{15}}$.
0
votes
1answer
15 views

Irreducible representation restricted to index 2 subgroup

Suppose $G$ is a (not nec. finite) group with index 2 subgroup $H$ and $k$ is a field (possibly of positive characteristic). Suppose $$\rho:G\to\mathrm{GL}_2(k)$$ is an irreducible 2-dimensional ...
0
votes
2answers
66 views

Does there exist a Vector that can't be written as a Tuple of Scalars?

The most abstract/general definition of a vector The most general definition of a vector is as an element of a vector space. Given a vector $u$, we can always say that there exists a vector space $V$ ...
1
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0answers
27 views

About decompositions of induced characters

Suppose $G$ is a finite group, $H\leqslant G$ is a subgroup. $\chi_1,...,\chi_s$ are all the irreducible characters of $G$ and $\psi$ is an irreducible character of $H$. Prove that if ...
2
votes
1answer
37 views

Each proper ideal is a product of prime ideals

$R$ is a commutative ring with unity. If $R$ is P.I.D. I want to show that each of its proper ideal is written as a product of prime ideals. $$$$ Since $R$ is a P.I.D. every ideal is a prime ...
5
votes
4answers
59 views

Total number of integers relatively prime to $p^2$

I am reading my number theory textbook and it states without proof that the total number of elements relatively prime to $p^2$ for some prime $p$ is $p(p-1)$. Why is this so? I know that the number of ...
0
votes
1answer
25 views

Irreducibility of a Polynomial after a substitution

I am trying to determine whether the polynomial $f(x) = x^6 + 34x^4 + 4x^2 + 89 \in \mathbb{Z}[x]$ is irreducible over $\mathbb{Z}$. Eisenstein's criterion doesn't help and I suppose I could determine ...
1
vote
1answer
23 views

Formal product of cycles in a permutation group

In Dixon's book Permutation Groups, there is a sentence saying that in a symmetric group $Sym(\Omega)$, "the second common way to specify a permutation is to write $x$ as a product of disjoint ...
0
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0answers
22 views

references of modular representations for finite group

What is modular representation for finite groups? I tried to find a book to understanding that but I could not find a good one. Are there any useful references?
-1
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1answer
27 views

Are these 2 groups isomorphic? [on hold]

I am struggling to find out whether these 2 groups are isomorphic or not. In general what should be the steps to find out? $$G = (\mathbb{Z}_6,+), \quad H = (\mathbb{Z}_2\times \mathbb{Z}_3, +)$$
1
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1answer
39 views

A question about tensor product and multiplication?

Let $k$ be a commutative ring, $A$ be an algebra over $k$. The tensor product $A\otimes A$ is over $k$. If $\sum_{i} a_i \otimes_k b_i=\sum_j c_j\otimes_k d_j$, where $a_i,b_i,c_j,d_j\in A$, I wonder ...
-2
votes
0answers
10 views

If $R$ is the ring of all rational numbers with odd denominators, $J(R)$ consists of all rational numbers with odd denominator and even numerator [on hold]

If $R$ is the ring of all rational numbers with odd denominators, then prove $J(R)$ consists of all rational numbers with odd denominator and even numerator.
1
vote
2answers
33 views

Let $M$ be a non zero maximal ideal in $\mathbb C[x].$ Prove that there exists $a\in \mathbb C$ such that $M=\langle x-a\rangle$.

Let $M$ be a non zero maximal ideal in $\mathbb C[x].$ Prove that there exists $a\in \mathbb C$ such that $M=\langle x-a\rangle$ (ideal generated by $x-a$). Attempt: As $\mathbb C[x]$ is a PID, ...
0
votes
0answers
21 views

If $R$ is a ring and $M$ is a left simple $R$-module, then $R/ann_{R}M$ is a left primitive ring

I'm attempting to prove that if $R$ is a ring and $M$ is a left simple $R$-module, then $R_1=R/ann_{R}M$ is a left primitive ring. I know that this becomes trivial if M is a faithful simple left ...
0
votes
1answer
29 views

Cardinality of Infinite Symmetric group [on hold]

How to show $|\mathrm{Sym}(\Gamma )|=2^{|\Gamma |}$ if $|\Gamma |$ is infinite?
-1
votes
0answers
12 views

A Nonzero Idempotent May be Quasi-regular

I am attempting to prove that the radical $J(R)$ doesn't contain nonzero idempotents but that a nonzero idempotent could be left quasi-regular. I know that an R with identity is left quasi-regular ...
2
votes
2answers
19 views

Finite dimensional representations of the Weyl algebra in characteristic $p>0$

I'm working through representation theory course notes of P. Etingof. In problem 1.26 it is asked to find all finite dimensional irreducible representations of the algebra ...
0
votes
3answers
25 views

Annihilator of rings [duplicate]

If $A$ is an $R$-module, I am having difficulty proving that $A$ is also a well-defined $R/ann(A)$-module with $(r+ann(A))a=ra$.
-1
votes
0answers
35 views

Analysis and algebra [on hold]

I'd like to know if there exist a field of the theoretical math that really combines analysis and algebra. Some people say that Model theory combines those two subjects but I personally want ...
2
votes
1answer
66 views
+50

Proof of algebraic set involving dimension

I need some help to understand the following proof. Let $k$ a field and $V$ an algebraic set. I note $\mathfrak{m}_P$ the ideal generated by $X_1-a_1,\dots ,X_n-a_n$ in $k[V]=k[X_1,\dots ,X_n]/I(V)$ ...