Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [abstract-algebra]

For questions about monoids, groups, rings, modules, fields, vector spaces, algebras over fields, various types of lattices, and other such algebraic objects. Associate with related tags like [group-theory], [ring-theory], [modules], etc. as necessary to clarify which topic of abstract algebra is most related to your question and help other users when searching.

Abstract algebra is the study of algebraic objects, i.e. sets endowed with one or more operations on the elements of those sets. In particular, the study of abstract algebra considers the algebraic structures and properties of which such operations induce. It can be considered as the generalization of the study of the algebraic structure of the integers and real numbers (arithmetic), or the study of matrices and vector spaces (linear algebra).

Some algebraic objects are monoids, groups, rings, fields, vector spaces, modules, algebras, and categories, among many other less prominent objects.

Examples

  1. The set of non-negative integers $\mathbb{N} = \{0,1,2,3,\dotsc\}$ is a monoid under the operation $+$.

  2. The integers $\mathbb{Z} = \{\dotsc,-1,0,1,\dotsc\}$ under the binary operation of $+$ form a group.

  3. Furthermore, $\mathbb{Z}$ has the structure of a ring when you consider it as being equipped with both addition and multiplication.

  4. The real numbers $\mathbb{R}$ with their usual addition and multiplication form a field.

  5. The set of $n\times n$ matrices with entries in $\mathbb{R}$ with matrix addition and multiplication form a ring.

  6. The set of $1\times n$ vectors over the real numbers, with vector addition, and multiplication by elements of the $n\times n$ real matrices on the right are an example of a module for the ring of matrices.

In addition to studying the objects themselves, abstract algebra considers homomorphisms between the objects and various constructions and tools, which are useful for studying the objects.

85022 questions
3
votes
2 answers

Groups and matrices

Let $K$ be the additive group of $\mathbb Z\oplus \mathbb Z$. If $A = \left(\begin{array}{cc} a & b\\ c & d \end{array}\right)$ is an $2\times 2$ matrix where $a, b, c, d$, are in $\mathbb Z$, then $HA*=\langle(a,b), (c,d)\rangle$ is the subgroup…
Mark Rutherford
  • 139
3
votes
2 answers

Galois connection from binary relations

Following picture is from Ordered Sets and Complete Lattices by Hilary A. Priestley. My question is how to prove part (ii)? I know that when $F_R=F$ it follows that $G_R=G$. So all is needed is to show that $F_R(A) = F(A)$, for every $A \subseteq…
onm
  • 71
3
votes
2 answers

What is the center of $End_A(S)$?

$k$ is a field and $A$ a finite-dimensional algebra over $k$, $S$ is a simple $A$-module. I know that $End_A(S) \subseteq End_k(S)$ and $End_k(S)$ has a center isomorphic to $k$. What is the center of $End_A(S)$? It contains $k$, for sure. This is a…
scsnm
  • 1,283
3
votes
2 answers

Semi direct product-groups that are isomorphic

The questions asks me to find all groups up to isomorphism of the semi direct product $C_5 \rtimes C_4$ Now I've done the working out to get four groups (Note I've used $X$ as an element of $C_5$ and $Y$ as an element of $C_4$ so in all of these…
Lolwat
  • 123
  • 7
3
votes
4 answers

Prove the map $\Phi:G\to\operatorname{Aut}G$ with $g\mapsto (x\mapsto g^{-1}xg)$ is an homomorphism.

According to this article, A group homomorphism is a map $f:G \rightarrow H$ between two groups such that the operation is preserved: $f(g_{1}g_{2}) = f(g_{1})f(g_{2})$ for all $g_{1}, g_{2} \in G$, where the product on the left-hand side is in…
NasuSama
  • 3,364
3
votes
3 answers

Find with proof the number of units in the ring $R=\mathbb{Z}_8 \times \mathbb{Z}_9 \times \mathbb{Z}_5$

Find with proof the number of units in the ring $$R=\mathbb{Z}_8 \times \mathbb{Z}_9 \times \mathbb{Z}_5$$ Since $\gcd(5,8,9)=1$, by Chinese Remainder Theorem, I get $R=\mathbb{Z}_{360}$ Since $\mathbb{Z}_{360}$ is not a field, not all elements has…
Idonknow
  • 15,643
3
votes
3 answers

Eisenstein criterion on f(x+1)

I need to show that the polynomial $f(x)=x^6+x^5+x^4+x^3+x^2+x+1$ is irreducible in $\mathbb{Z}[X]$ and in $\mathbb{F}_2[X]$. As we can't find a prime number p satisfying the conditions for the Eisenstein criterion, I did not know how to solve it. I…
Mikalo
  • 123
3
votes
1 answer

How to show there are no simple groups of order 760 using Sylow's theorem

Prove that there does not exist a simple group of order 760. I was trying to solve it by using Sylow's Theorem but I am unable to prove it. Thanks for any help.
Ester
  • 3,057
  • 2
  • 23
  • 34
3
votes
2 answers

Is an algebraic structure a mathematical structure?

I am trying to get a sense of the relationships between various abstract math concepts. I keep seeing mathematical structures and algebraic structures mentioned and explained, but I never see them mentioned together. Any reason for that? Based on…
Erik Engheim
  • 183
3
votes
3 answers

Relatively prime polynomials in $k[x]$

suppose $k$ is a field and take two polynomials $p(x),g(x) \in k[x]$. If $(p,p')=1$ and $(g,g')=1$ then why is it true that $(pg,(pg)')=1$ where $'$ denotes formal derivative and $(,)$ denotes gcd (i.e (pg)' denotes the derivative of pg). I tried…
user18
  • 31
3
votes
2 answers

Disprove: If $H$ and $K$ are subgroups of a group $G$, then $HK$ is a subgroup of $G$

I know this statement is true when $G$ is not abelian. But I'm struggling to find a counterexample. I've only considered $S_{4}$ with subgroups $D_{4}$ and $A_{4}$. I'm at a very basic (less than a month old) level of abstract algebra, so I don't…
GrainyDarkPurple
  • 41
3
votes
2 answers

What does it mean for a group to be a quotient of another and how does it imply isomorphism?

I was reading a proof of the following proposition and had one small doubt about the proof: Proposition: Let $H$ be a subgroup of $G,$ and $\mathcal{A}$ a $(\text {sub})$ normal series in $G .$ Then the series $$ \mathcal{A}_{H}: E=A_{0} \cap H…
PCeltide
  • 1,534
3
votes
4 answers

Any quotient group of $(\mathbb Q,+)$ is torsion

We know that $\mathbb Q/\mathbb Z$ is a torsion group. Can we extend this to every quotient group of $\mathbb Q$? Thanks for hints.
Baker
  • 71
3
votes
1 answer

Index of $n\mathbb{Z}$ in $\mathbb{Z}$.

This is problem 1 of Ch.2, Section 6 in Artin's Algebra (Ed.1). It asks to determine the index $[\mathbb{Z} : n\mathbb{Z}]$. I have that if $|n|\geq 1$, then $[\mathbb{Z} : n\mathbb{Z}] = |n|$, but if $n=0$, then $[\mathbb{Z} : n\mathbb{Z}] =…
user59083
3
votes
1 answer

Showing every $p$-subgroup of order $p^i$ is in a subgroup of order $p^{i+1}$

Let $k$ be the largest positive integer such that $p^k \mid |G|$ and let $|H|=p^i$ where $0
user10444
  • 2,846
1 2 3
…
99 100