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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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Quotient group $\mathbb{Q}/\mathbb{Z}$

How do I prove that the quotient group $\mathbb{Q}/\mathbb{Z}$ is a group? What is its unity element? How do I prove that all elements are of finite order?
charles_xavier
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Unique maximal ideal in an algebra

Let $Z$ be an algebra over a field $K$ such that $Z$ is generated by a single nilpotent element. Why is $Z$ a local ring? Would this follow from the Chinese Remainder theorem? Let $m$ be a maximal ideal of $Z$ then since $Z$ is Artinian then $Z…
Curtier
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Let $G$ be a group. Let $a, b, c$ denote elements of $G$, and let $e$ be the neutral element of $G$. Prove the following.

The problems are quoted. They are followed by proofs. Please, check my work. If $ab = e$, then $ba = e.$ By hypothesis, $a = b^{-1}$ and $b = a^{-1}.$ Then, $b^{-1}a^{-1} = e.$ So, $bb^{-1}a^{-1}a = bea.$ Thus, $e = ba.$ If $abc = e$, then $cab…
AbstraktAlgebra
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Prove or disprove by counterexample that in every group, every element has a square root

For every $x \in G$, there is some $y \in G$ such that $x=y^2$. (This is the same as saying that every element of G has a square root) Now, I'm not sure but I've been trying to think of counter-examples and I thought of the group of integers under…
mika
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Sum of submodules A and B is the smallest module that contains A and B

From Dummit and Foot, Abstract Algebra ch10.2: Defn: Let A,B be submodules of the R-module M. The sum of A and B is the set $A+B=\{a+b| a\in A, b\in B \}$ it is easily checked that the sum of two submodules A and B is a submodule and is the smallest…
Edison
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Groups and Symmetries

Let $G$ be a group and let $f:G\to G$ be an isomorphism. Let $H = \{ a \in G \; | \; f(a)=a^{-1}\}$ Prove that $H$ is a subgroup of $G$ if and only if $G$ is abelian. My attempt: I already know that $H$ is a subset of $G$ since $a$ is in $G$. And…
Jeffrey
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Why are equivalent conditions equal for normalizer

Hello so I need to proof that Center is subgroup of normalizer which I did but I did it using the following definition of $N_G(A) = \{h \in G : hA = Ah\}$. But the definition that I was given is $N_G(A) = \{ h \in G : hah^{-1} \in A \;\forall a \in…
user111750
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Is there a general way to find what $Aut(C_n)$ Is Isomorphic to?

I'm asked to describe $Aut(C_{21}),Aut(C_{24})...$ as a product of cyclic groups - but I'm wondering is there a general way to do this?
automobile1
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If $G$ is an abelian group, is then $H=\{a\in G: a*a=a\}\leq G$?

I feel very silly for asking this question: Let $G$ be an abelian group. Show that $H=\{a\in G:a*a=a\}$ is a subgroup of $G$. I did not get this question right, but this is what I managed to observe: Notice…
wjmolina
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quick question on Galois theory

I am reading a book on Galois theory and, as per usual (why is that?), all sorts of unproven properties start to magically appear in the proofs of the couple of theorems that really matter. The author tries to prove that, when $L:K$ is a finite…
Romanda de Gore
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Finding the order of all the elements in Group $\mathbb{Z}_{12}$

I know that the order of an element $a$ in a group $G$ is the smallest positive integer $m$ such that $a^m=e$ and so for $(\mathbb{Z}_{12},+)$ we have $[0]$ is the identity of order 1. $[1]$ is order 12 because…
alkabary
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finite/algebraic field extensions and minimal polynomial

1) To show: $L/K$ field extension is algebraic iff every subring with $K\subset R\subset L$ is a field. My answer: I can write $R=\bigcup\limits_{\alpha\in R}K[\alpha]$ with $K[\alpha]$ (field as $L/K$ algebraic) is the image of $f\mapsto…
user208604
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group of order 21 is abelian or not

Let $G$ be a group of order 21 .Question is whether it is abelian or not? Number of Sylow 3 subgroups =1+3k divides 7 hence k=0 or k=2.Number of Sylow 7 subgroups=1+7k divides 3 hence k=0.In both cases if we get a unique sylow 3 and a unique sylow…
Learnmore
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Let I and J be ideals of a ring R. How to prove that IJ is closed under multiplication?

Let I and J be ideals of a ring R (not necessarily unitary). I need to prove that IJ is an ideal of R contained in $I\cap J$. I can prove all the other properties of ideal. However, I cannot prove that IJ is closed under multiplication. I think I…
user10024395
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Extension of the factor theorem

Motivation If you don't care the least bit about motivation, scroll down. The following is a standard result in a first algebra course: Factor Theorem. Let $R$ be an integral domain and $p\in R[X]$ with a root $a \in R$. Then there exists a…
kahen
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