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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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Bijective functions on a $G$-set when $X$ is transitive

Let $X$ be a nonempty $G$-set. Show that if $X$ is transitive (i.e., for all $x, y$ on $X$, there is a $g$ on $G$ such that $gx=y$) Then every function $f: X\to X$ is a bijection. There is something else, $f$ holds for the following property $f(gx)…
Pokemon master
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Find a homomorphism phi from U(40) toU(40) kernal{1,9,17,33} and phi (11)=11

Find a homomorphism $phi$ from $U(40)$ to $U(40)$ with kernel ${1,9,17,33}$ and $phi(11)=11$ Iam looking for a solution since I have already tried several times myself.
Malcolm
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Why is the rank of a finitely generated finite abelian group zero?

I haven't yet proven the fundamental theorem of finitely generated abelian groups, but it is stated without proof in my textbook, Abstract Algebra by Dummit and Foote, page 159; its proof appears in a later chapter. The textbook also states that the…
user140374
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How to prove that it is a group?

Let $G$ be a set with a binary operation *, associating to each pair of elements $x$ and $y$ of $G$ a third element $x*y$ of $G$. Suppose that the following properties are satisfied: $(x*y)*z = x*(y*z)$ for all elements $x$, $y$, and $z$ of $G$…
velut luna
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Prove that $f(x)=x^2$ is a homomorphism of ${G}$ onto $ {H}$

Let G be an abelian group. Let $H=\{x^2 : x \in G\}$ and $K=\{x \in G: x^2=e\}$ I know I have to use the fundamental homomorphism theorem. And I also know if $f$ is a homomorphism from $G$ onto $H$ then $H$ is isomorphic to the quotient group of $G$…
Bradley
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How do i prove that $\gcd(s,n)=\gcd(t,n) \Rightarrow \langle a^s\rangle =\langle a^t\rangle$?

Let $G$ be a finite cyclic group generated by $x$. $(|G|=k)$ Let $n,m\in\mathbb{Z}$ such that $\gcd(n,k)=\gcd(m,k)$. Then, $\langle x^n\rangle=\langle x^m\rangle$. I can prove the converse, but i don't know how to prove this one. I can show that…
KetLSD
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Show that if $N$ is a normal subgroup of $G$ which contains all commuters then $G/N$ is abelian.

I am working on my proof for class and I was wondering if this look ok? Let $N$ be a normal subgroup of $G$ we want to show that $G/N$ is abelian, or $(aN)(aN) = abN = baN = (bN)(aN)$. Since $N$ contains all commutors, then let $aba^{-1}b^{-1}N =…
spitfiredd
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Why is the group G a normal subgroup of itself?

Apparently $g^{-1}Gg=G$ for all $g$ in $G$. I understand that by closure if you multiply on the right by $g$ and the left by $g^{-1}$ you will get an element of the group $G$. However, how do i know that some of the elements of $g^{-1}Gg$ will not…
usainlightning
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Definition of field extension

One usually defines field extension $E/F$ whenever $F\subseteq E$. However, few authors would define field extension $F/K$ whenever there is a nonzero field homomorphism $F\rightarrow E$ (see e.g Definition 3.1 page 35…
user136993
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Consider group of permutation matrices and write out elements isomorphic to the group and exhibit it

Consider the group of permutation matrices $G =\{I_3, P_1, P_2, P_3, P_4, P_5\}$ For $n=3$ the permutation matrices are $I_3$ and the five matrices are: \begin{equation*} P_1 = [1,0,0;0,0,1;0,1,0] \\ P_2 = [0,1,0;1,0,0;0,0,1] \\ P_3 =…
Jessica
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Group Isomorphisms

I'm having difficulties with the underlined parts of the solution.
usainlightning
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Elements of alternating group $A_3$

List all the elements of the alternating group $A_3$ written in cyclic notation. I come up with Identity $(1)$ Obviously $(123)$
Jessica
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Ring homomorphisms

Let $R$ and $S$ be rings. If $f : R \rightarrow S$ is a surjective ring homomorphism, then $$\bar{f} : R / \ker(f) \longrightarrow S, x + \ker(f) \longmapsto f(x)$$ is an isomorphism. First, I wanted to show that $\ker(f)$ is an ideal. I let $r…
Cookie
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Problem on polynomial in several variable

Let $A$ be a commutative ring and let $R= A[x_{1},...,x_{n}]$. Let $a \in A$ and $f \in R$ satisfy $f(x_{1},..x_{n-1},a) = 0$ (as a polynomial in the variables $x_1,\ldots,x_{n-1}$). How do we show that $f= (x_{n} - a) g$ for some $g \in R$?
user92360
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Abstract algebra and the knight. (as in chess, not the knave that the box suggests) - How do I study the patterns?

(background removed -too long, note terminology may be wrong, I had to teach myself, armed with a 4AA powered chess computer) When moving a knight in chess its colour changes the colour of the square on which the knight rests changes for each…
Alec Teal
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