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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
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The presentation of the dihedral group $D_{2n}$

We know that $D_{2n}= \langle r,s \mid r^{n}=s^{2}=1, rs=sr^{-1} \rangle.$ From Dummit & Foote, any other relations of $r,s$ can be deduced from the relations given in the presentation. The book claims that this is true because we can tell whether…
Keith
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Group of translations is a normal subgroup of group $E(2)$ of isometries of $\mathbb{R}^2$?

I'm trying to see that: The group of translations $T=\{t(x)=x+a : a \in \mathbb{R}^2 \}$ is a normal subgroup of group $E(2)$ of isometries of $\mathbb{R}^2$. I know the definitions of a normal subgroup, but I don't understand how the translations…
mavavilj
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What's group $E(2)$ of isometries of $\mathbb{R}^2$?

I'm trying to prove normal subgroups of the group $E(2)$, but I haven't been given, what the group $E(2)$ of isometries of $\mathbb{R}^2$ is like. What is it like?
mavavilj
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Anticommutative operation on a set with more than one element is not commutative and has no identity element?

An anticommutative operation on a set X to be a function $\cdot:X\times X\rightarrow X$ satisfying two properties: (i) Existence of right identity: $\exists r\in X:x\cdot r=x$ for all $x\in X$ (ii) $x\cdot y=r\iff(x\cdot y)\cdot(y\cdot x)=r\iff x=y$…
Timothy
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An Abelian group of order n is cyclic

In the link below, someone has made a comment that an abelian group of order $n$ with $n=p_1...p_n$, where $p_1...p_n$ are primes, is cyclic. Can someone please explain why this is the case? Thanks in advance! An abelian group of order $n$ with…
Frank Booth
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Monoid with two binary operations

I have a set of functions $S$ whose domains are of the form $[0,x]\subset \mathbb{R}$. I then have two binary operation $(S,+,-)$. The "$+$" operation gives a semigroup structure: take $f,g\in S$ with $dom(f)=[0,x_f]$ and $dom(g)=[0,x_g]$…
Brian Paden
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Is a Galois extension always a normal extension?

I have been told that a Galois extension is always a normal extension. The proof goes like below: $F \subset K$ is a Galois extension. Let $g(x)$ be an irreducible polynomial in $F[x]$ and $\beta$ be a root of $g(x)$. $G(K/F)$ acts on the root of…
Keith
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Automorphisms and Finite Cyclic Groups

The Question: Let C be a finite cylic group and let $ E \subset C $ be a subgroup of C. Prove that every automorphism $\alpha: C\rightarrow C$ we have $\alpha(E) = E$. I know that every subgroup of a cylic group is cylic, and that for any $e \in E$…
IntegrateThis
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Let $p$ be a nonzero prime element of an integral domain D. Show that $p$ is irreducible.

Let $p$ be a nonzero prime element of an integral domain D. This means that whenever $p$ divides a product $ab$ with $a, b \in D$, it must divide $a$ or $b$. Show that $p$ is irreducible. I tried to solve this question by assuming p is reducible,…
Nhay
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Show that there exists a ring isomorphism $R/(I\cap J) \cong R/I \times R/J$

Let $R$ be a ring and $I$ and $J$ be ideals in $R$ such that $I + J = R$. Show that there exists a ring isomorphism $R/(I\cap J) \cong R/I \times R/J$. I've already proved that $IJ= I\cap J$. Feels like I am pretty closed to the answer, but trapped.…
J.doe
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Prove that $S_3 \times \mathbb{Z/2Z}$ is isomorphic to $D_6$. Can you make a conjecture about $D_{2n}$? Prove that conjecture.

I already have shown that product $S_3 \times \mathbb{Z/2Z}$ is isomorphic to $S_6$, by taking the subgroups $H = \{id, r^3\}$ and $K = \{id, r^2,r^4,s,sr^2,sr^4\}$ and multiplying $KH$ to get the group $D_6$. I do not know what I can conjecture in…
J. Medley
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Proof of Division Algorithm in $F[x]$

Division Algorithm for $F[x]$ Theorem: Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ and $g(x)=b_mx^m+b_{m-1}x^{m-1}+\cdots+b_0$ be two elements of $F[x]$ with $a_n$ and $b_m$ both nonzero of $F$ and $m>0$. Then there are unique polynomials $q(x)$ and…
H. Vigilia
  • 83
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The norm of a Gaussian prime is a prime or a square of a prime in $\mathbb{Z}$

Let $a \in \mathbb{Z}[i]$ be a prime element. Show that $N(a)$ is a prime or a square of a prime. I know that the converse is true. That is, if the norm of $a$ is prime then $a$ is a Gaussian prime. But how do I prove the claim above? Thanks!
user112358
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Proof of $+0 = -0$

How do you prove $+ 0 = - 0$ ? I have no clue where to start from. (I am a 11th Grader). Can it be done only using concepts I have learned till now or will I need some more concepts?
SS_C4
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How many elements of order 7 in a Non-simple group of order 168?

If the group is simple, sylow is enough, otherwise i have no idea.
Kai
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