Abstract algebra is a study of algebraic objects. Some of the more common algebraic objects are Groups, Rings, Fields, Vector spaces, Modules, and other advanced topics.
1
vote
2answers
21 views
Generalized Rationalization in Finite Radical Field Extensions
In the square root case of a radical extension of, say, $\mathbb{Q}$, we have that $\mathbb{Q}(\sqrt{2}) = \{a + b \sqrt{2} | a, b \in \mathbb{Q} \}$.
The only semi-hard axiom to prove is that ...
2
votes
1answer
20 views
Unique largest normal pi-subgroup
Let $\pi$ be a set of prime numbers. A finite group is said to be a $\pi$-group if every prime that divides its order lies in $\pi$. If $G$ is finite, show that $G$ has a unique largest normal ...
2
votes
0answers
29 views
$\gcd(|G:H|,|G:K|)=1$ implies $HK=G$ [duplicate]
Let $G$ be finite and assume $H,K$ are subgroups of $G$ with $\gcd(|G:H|,|G:K|)=1$. Show that $HK=G$.
What I did: I'm using a result that $|K:H\cap K|\leq |G:H|$ with equality iff $HK=G$. So ...
1
vote
1answer
29 views
The sum of a homomorphism that sends $M$ to $\mathfrak aN$ and a surjection is an isomorphim between finitely generated modules $M$ and $N$
$R$ is a commutative ring, $M$ and $N$ finitely generated $R$-modules, $\alpha, \beta\in \operatorname{Hom}_{R}(M,N)$, $\mathfrak a\subset \operatorname{rad}(R)$ and $\alpha $ is surjective while ...
0
votes
1answer
30 views
Group equals union of three subgroups
Suppose $G$ is finite and $G=H\cup K\cup L$ for proper subgroups $H,K,L$. Show that $|G:H|=|G:K|=|G:L|=2$.
What I did: so if some of $H,K,L$ is contained in another, then we have $G$ being a union of ...
3
votes
2answers
36 views
Frattini subgroup is set of nongenerators
The Frattini subgroup $\Phi(G)$ is the intersection of all maximal subgroups of $G$. (If there are none, then $\Phi(G)=G$.) We say that an element $g\in G$ is a nongenerator if whenever $\langle X\cup ...
4
votes
1answer
40 views
Linear transformation invertible or not?
Let $ T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a linear transformation defined such that the inner product of $\langle T(v), v \rangle = 0$ for all $v$ in $\mathbb{R}^2$. Is $T$ invertible or ...
0
votes
3answers
40 views
Given two sets, finding two non trivial homomorphisms that are not isomorphisms
Is it possible to have two non trivial homomorphisms that are not isomorphisms for given two Groups?
I am specially interested in additive/remainder Group of Integers and multiplicative (arithmetic ...
0
votes
1answer
30 views
Prove that if $ h \circ f = g $ then $ h $ is an $A$-algebra homomorphism.
Let $f:A\rightarrow B,\ g:A\rightarrow C$ be ring homomorphisms. An $A$-algebra homomorphism $h:B\rightarrow C$ is a ring homomorphism which is also an $A$-module homomorphism.
Please prove that if ...
0
votes
0answers
30 views
Conditions for a ring to be isomorphic to the product of rings.
Let $R,R_1,\dots,R_n$ be rings. Show that $R\cong R_1 \times \cdots\times R_n$ if and only if there exist ideals $I_1,\dots,I_n$ of $R$ such that
(a) $I_i\cong R_i$ for all $i$
(b) $R= I_1 ...
0
votes
1answer
32 views
Matrix ring over a field and its ideals
Let $M_n(R)$ be the matrix ring over a commutative ring $R$ and let $I$ be an ideal of $R$.
1) Show that if $R=F$ is a field then the only nonzero ideal of $M_n(F)$ is $M_n(F)$ itself.
2) Let ...
1
vote
1answer
43 views
What is the factor group $\mathbb{Z}/5\mathbb Z$?
I am trying to understand the concept of factor group. The definition of factor group I know is the following: Let $G$ be a group and $H$ be a subgroup of $G$. Then the group of cosets denoted by ...
2
votes
2answers
49 views
Listing subgroups of a group
I made a program to list all the subgroups of any group and I came up with satisfactory result for $\operatorname{Symmetric Group}[3]$ as
...
3
votes
3answers
33 views
Subrings and homomorphisms of unitary rings
Let $(R,+,\cdot)$ be a ring (in the definition i use multiplication is associative operation and it's not assumed that there is unity in the ring).
I've seen two definitons of subring.
1) non-empty ...
3
votes
0answers
37 views
Units of the quotient of an order
Let $n$ be a positive integer and $R$ be an order in a imaginary quadratic number field such that $disc(R)$ is prime to $n$. Further suppose that for every prime $p$ dividing $n$, $p$ is inert in $R$. ...
1
vote
1answer
34 views
Non-self-mapping automorphism implies abelian [duplicate]
Suppose $\sigma\in\text{Aut}(G)$. If $\sigma^2=1$ and $x^{\sigma}\neq x$ for $1\neq x\in G$, show that if $G$ is finite, it must be abelian.
There's a hint to show that the set ...
3
votes
1answer
46 views
Set of homomorphisms form a group?
Given vector spaces $V, W$ over field $F$, the set of all linear maps $V \to W$ forms a vector space over $F$ under pointwise addition.
Is there an analogue for groups? Can the set of all ...
7
votes
2answers
80 views
If $f\colon G\to H$ is a surjective homomorphism, then $|C_G(g)| \geq |C_H(f(g))|$
Let $G$ be finite, $f\colon G\to H$ be a surjective homomorphism (hence $H$ is finite) and $g \in G$. Prove the order of center of $g$ in $G$ is greater than or equal to the order of the center of ...
2
votes
1answer
36 views
Group equals union of two subgroups [duplicate]
Suppose $G=H\cup K$, where $H$ and $K$ are subgroups. Show that either $H=G$ or $K=G$.
What I did: For finite $G$, if $H\neq G$ and $K\neq G$, then $|H|,|K|\le |G|/2$. But they clearly share the ...
2
votes
1answer
40 views
What is the exact definition of polynomial functions?
I'm trying to understand the difference between polynomial functions and the evaluation homomorphisms. I noticed that I don't know what's the exact definition of a polynomial function, although I've ...
4
votes
0answers
29 views
Find all the central extensions of $\Bbb{Z}_2$ by $\Bbb{Z}_2 \times \Bbb{Z}_2$.
Find all the central extensions of $\Bbb{Z}_2$ by $\Bbb{Z}_2 \times \Bbb{Z}_2$.
At first I was going to say that, since central extensions mean a trivial homomorphism, there must be only one: ...
1
vote
1answer
24 views
Group of mapping is a subset of symmetric group
Let $G$ be a group of mappings on a set $X$ with respect to function composition. Show that if $G$ contains some injective function, then $G\subseteq \text{Sym}(X)$.
What I did: If $X$ is finite, ...
11
votes
2answers
110 views
Prove that $\exists a,g,h\in G\colon h=aga^{-1}, g\neq h ,gh=hg$ in a finite non-abelian group $G$.
Let $G$ be a finite and non-abelian group. How do I prove the following statement? $$\exists a,g,h\in G \colon\quad h=aga^{-1},\ g\neq h ,\ gh=hg.$$
Thanks in advance.
6
votes
1answer
49 views
Quaternion group associativity
Consider the eight objects $\pm 1, \pm i, \pm j, \pm k$ with multiplication rules:
$ij=k,jk=i,ki=j,ji=-k,kj=-i,ik=-j,i^2=j^2=k^2=-1$,
where the minus signs behave as expected and $1$ and $-1$ ...
6
votes
2answers
61 views
$V=\{A\in M_{3\times 3}(\mathbb{R}):\text{trace}(A)=0\}$ is isomorphic to $\text{span}\{AB-BA:A,B\in V\}$
Background: Let
$$V:=\{A\in M_{3\times 3}(\mathbb{R}):\text{trace}(A)=0\}$$
be the vector space of $3\times 3$ real matrices with vanishing trace, and let $[\cdot,\cdot]:V\times V\to V$ be defined by
...
1
vote
0answers
29 views
$\mathbb{Z}/N\mathbb{Z}$-algebra
Let $N>0$ be an integer and $A$ be a commutative free algebra of rank $2$ over $\mathbb{Z}/N\mathbb{Z}$. Is it true that $A$ is isomorphic to $(\mathbb{Z}/N\mathbb{Z}[X])/(P)$ where $P \in ...
0
votes
0answers
31 views
Prove any two cycles in Perm(x) of the same length are conjugate?
Not really sure how to go about a proof such as this. Any help would be appreciated.
Some Lemma's from my notes which I think I'm supposed to use:
Lemma 1: Conjugating a cycle with any permutation ...
4
votes
2answers
45 views
Can the concept of field extensions be applied equally well to UFDs?
In a nutshell, a field extension is where you take a polynomial $p(x)$ that is irreducible in some field $F$, then define $\alpha$ as a root of $p$, then add $\alpha$ to $F$, then add the minimum ...
1
vote
3answers
54 views
$R$ is an integral domain iff $R[x_1,…,x_n]$ is a integral domain
I would like to prove that $R$ is an integral domain iff $R[x_1,...,x_n]$ is a integral domain. The converse is trivial, since R can be viewed as a subring of $R[x_1,...,x_n]$. In order to prove the ...
3
votes
2answers
44 views
Jacobson Radical and Finite Dimensional Algebra
In general, it is usually not the case that for a ring $R$, the Jacobson radical of $R$ has to be equal to the intersection of the maximal ideals of $R$.
However, what I do like to know is, if we are ...
1
vote
2answers
82 views
There are infinitely many choices of $(\alpha_1,\dots,\alpha_n)$ such that $f(\alpha_1,\dots,\alpha_n)\neq 0$
I'm trying to solve this exercise in the page 10 of this book
Maybe I'm forgetting something, but I couldn't solve this exercise, I need a hint or something to begin to solve this question.
Thanks ...
0
votes
0answers
23 views
Expressing a Sequence as a Function of n (Cartan Groups)
The problem is concerning a variation of A141419,
the only difference is that my sequence, instead of being like shown on OEIS:
{1},
{2, 3},
{3, 5, 6},
{4, 7, 9, 10},
{5, 9, 12, 14, 15},
{6, 11, 15, ...
6
votes
0answers
63 views
Degree of transitive constituents is odd implies $|G|$ is odd
I want to prove that: the order of a permutation group $G \le S^\Omega$ is odd if and only if the degrees of all transitive constituents of $G$ and the degrees of all transitive constituents of each ...
2
votes
1answer
27 views
Lagrange's Theorem for further elementary consequences
Question: Let $G$ be a finite group, and let $H$ and $K$ be subgroup of $G$. Prove: suppose $H$ and $G$ are not equal, and both have order the same prime number $p$, Then $H\cap K=\{e\}$.
This is my ...
5
votes
1answer
56 views
Show that If k is odd, then $\Bbb{Q}_{4k}$ is isomorphic to $\Bbb{Z}_k \rtimes_{\alpha} \Bbb{Z}_4$ for some $\alpha$
Show that if k is odd, then $\Bbb{Q}_{4k}$ is isomorphic to $\Bbb{Z}_k \rtimes_{\alpha} \Bbb{Z}_4$ for some $\alpha: \Bbb{Z}_4 \rightarrow Aut(\Bbb{Z}_k)$. Calculate $\alpha$ explicitly.
We know ...
0
votes
0answers
20 views
Mapping on a set with respect to function composition
In Isaacs' Algebra, I found the following exercise
Let $G$ be a group of mappings on a set $X$ with respect to function composition. Find an example where $G$ is not a subset of $\text{Sym}(X)$ and ...
1
vote
0answers
29 views
Is there a “nice” description of the algebraic closure of the field of multivariable polynomials?
I'm interested in the ring $\mathbb{Z}[x_1, \dots, x_n]$ (multivariable polynomials with integer coefficients). Specifically, I want to know its algebraic closure (the set of roots of polynomials in ...
2
votes
1answer
24 views
Minimal polynomial matrix
I want to show that $ x^n-1$ is the minimal polynomial of the permutation matrix $P:=(e_2,e_3,....,e_n,e_1)$ where $e_i$ is the i-th unit vector written as a column vector.
And now I have to show ...
0
votes
2answers
40 views
tensor product and direct product of algebra presentations
Let $R$ be a commutative unital ring and $R\langle x_i\mid f_j\rangle$ denote a unital $R$-algebra presentation.
Q1: What is the presentation of $R\langle x_i\mid f_k\rangle\otimes R\langle y_j\mid ...
6
votes
1answer
31 views
Unity in the rings of matrices
Suppose we are given an arbitrary ring $R$. Then the set $M_n(R)$ of all square matrices with elements from $R$, together with usual matrix addition and multiplication forms a ring. If R is a unitary ...
2
votes
2answers
47 views
Commutator property
Can you please show me how to prove this? If H,K,L are normal subgroups of G then $$[[H,K],L]\subseteq [[K,L],H][[L,H],K]$$ Thanks in advance!
1
vote
0answers
58 views
How to solve it in radicals?
How to solve the equation $x^5+10x^3+20x-18=0$ in radicals? One of its roots is
$$\frac 1 5\, \left( -\frac1 4- \frac 1 4\,\sqrt {5}+\frac 1 4\,\sqrt {-10+2\,\sqrt {5}}
\right) \sqrt ...
2
votes
2answers
23 views
Calculating the centralizer of a matrix in a general linear group.
Let $G = GL(3,\mathbb{R})$ be the general linear group over the reals , of order $3$ , and let
$A\in G$ be :
$$
A=\begin{pmatrix}
-1 & 0 & 0 \\\
0 & 1 & 0 \\\
0 & 0 & 2
...
0
votes
0answers
21 views
GCD of polynomial in GF(2) and the reals
we were asked to calculate the gcd of $p=x^5+x^4+x^3+x^2+x+1$ and $q=x^4+x^3+x^2+x$ in the fields $\mathbb{R}$ and $GF(2)$
I first did $\frac{p}{q}=x$ with remainder $x+1$
then I did ...
0
votes
2answers
49 views
$Z(GL_n(\mathbb R)) = \{aI : a\neq0\} $ [duplicate]
$Z(GL_n(\mathbb R)) = \{aI : a\neq0\} $
This article is the general case for $GL(n,k)$ where $k$ is a field. Could I prove it only with a basic linear algebra?
0
votes
0answers
27 views
What does the general factorization of a multivariable polynomial look like?
Any polynomial $p$ in the ring $\mathbb{Z}[x]$ factors into an expression of the form $z_0(x - z_1) \dots (x - z_k)$, with $z_0, \dots, z_k \in \bar{\mathbb{Z}}$ (the algebraic closure of ...
2
votes
1answer
55 views
prove that a non-abelian group of order $10$ must have a subgroup of order $5$.
prove that a non-abelian group of order $10$ must have a subgroup of order $5$.
using Cauchy's theorem proof is easy but how can I do this without using this?
1
vote
0answers
29 views
Subfields $k\subseteq F\subseteq k(x_1,\dots,x_n)$ and their intersection with $k[x_1,\dots,x_n]$
By this thread, if I have a subfield $k\subseteq F\subseteq k(x_1,\dots,x_n)$, $F$ is of the form $F=k(\phi_1,\dots,\phi_m)$ for some rational functions $\phi_1,\dots,\phi_m\in k(x_1,\dots,x_n)$. But ...
5
votes
3answers
181 views
What are morphisms of functors
I am not been able to understand, what is a morphism between two functors. I have gone through the formal definition involving a commutative diagram. Can someone explain that to me in a bit more ...
2
votes
2answers
33 views
Smith-Normal Form
Could someone provide a good reference to look up the existence and uniqueness of Smith Normal Form (SNF) for a PID? I have seen it done for Euclidean domains but not for a PIDs. I know the difference ...
