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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.

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$G$ is Abelian if it has no element of order $2$ and $(ab)^2=(ba)^2$

Suppose that $G$ is a group that there exists no element $x \neq e$ such that $x^2=e$. Moreover, for every $a,b \in G$ we have $(ab)^2=(ba)^2$. Prove that $G$ is Abelian. Well, I attempted to prove that $(aba^{-1}b^{-1})^2=e$ because then if I could…
user66733
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Abstract Algebra Cosets

I don't know how to even approach this problem. Let G be the group of rotations of a plane about a point P in the plane. Thinking of G as a Group of permutations of the plane, describe the orbit of a point Q in the plane.
KGTW
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Does my proof make sense?

Theorem: For groups $(\Bbb R,+)$ and $(\Bbb R,*)$ (both only dealing with positive integers) there is a function $\phi$ that turns $(\Bbb R,+)\to(\Bbb R,*)$ and vice versa. Proof: Assume $(\Bbb R,+)\to(\Bbb R,*)$. So there is a function where…
Derek Marlon
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Difference between $k[X]$ and $k(X)$ where $k$ is a field

Can someone please explain me elaborately what is the difference between $k[X]$ and $k(X)$ where $k$ is a field?
humto
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Abelian group that is non-cyclic

Construct an abelian group of order 12 that is not cyclic. Can somebody please explain me with examples non-cyclic groups I'm having a hard time understanding.
Candy Pelagio
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Definition of left regular representation of $G$.

I don't really understand the definition of left regular representation of $G$. Could anyone give me some explanation or examples? The permutation representation afforded by left multiplication on the elements of $G$ (cosets of $H = 1$) is called…
Tumbleweed
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Tricky Algebraic Reduction

So I'm trying to work my way through Ernst Kummer's De Numeris Complexis, and I've reached a point where I keep stumbling over something that should be very, very simple. After almost an hour of playing around with this, I have been unable to solve…
StormyTeacup
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Order of elements in cyclic groups...

Let $G$ be a cyclic group of order $n$. Suppose $x$, $y$ are two elements of order $d$, where $d$ divides $n$. Show that $y = x^m$, where $m$ is an integer coprime to $n$. I know $y=x^m$ since the subgroups generated by $x$ and $y$ must be…
Johnny Apple
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Question about proof for $S_4 \cong V_4 \rtimes S_3$

In my book they give the following proof for $S_4 \cong V_4 \rtimes S_3$ : Let $j: S_3 \rightarrow S_4: p \mapsto \left( \begin{array}{cccc} 1 & 2 & 3 & 4 \\ p(1) & p(2) & p(3) & 4 \end{array} \right)$ Clearly, $j(S_3)$ is a subgroup $S_4$…
user12205
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The analogue of euler's theorem in $\mathbb{F}_p[x]$

Let $p$ be a prime number, and $m(x)$ be in $\mathbb{F}_p[x]$. the analogue of euler's theorem is that for certain polynomials $a(x)$ in $\mathbb{F}_p$, and some number $\phi_p(m)$, $$a(x)^{\phi_p(m)} = 1\mod m(x)$$ for which polynomials $a(x)$ does…
abbey
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Density of $\mathbb{Z}[\sqrt2]$

Show that between any two numbers in $\mathbb{Z[\sqrt2]}$, there is another number in $\mathbb{Z[\sqrt2]}$. {$(-1+\sqrt2)^1,(-1+\sqrt2)^2,...,(-1+\sqrt2)^n$} represents an infinite sequence of numbers in $\mathbb{Z[\sqrt2]}$ that approaches 0 from…
S. A. Lloyd
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What is this Weierstrass' proof of uniqueness of $\mathbb{R}$ and $\mathbb{C}$ algebras?

I'm reading Derbyshire's Unknown Quantity. It's an interesting exercise to enumerate and classify all possible algebras. Your results will depend on what you are willing to allow. The narrowest case is that of commutative, associative,…
Red Banana
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Thoughts that guide a proof in abstract algebra

I would like to know which strategies you use when proving abstract theorems. I have a particular example in mind. I worked some hours on a proof of the following: In the given diagram modules with corresponding module homomorphisms are shown. The…
Timm von Puttkamer
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Condition on monic second-order polynomials so that a commutative ring is an integral domain

I am trying to prove this result: Let $A$ be a commutative ring different from $\{0\}$, $\mathbb{Z}/4\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}[X]/(X^2)$. Prove that if any monic polynomial of degree $2$ of $A[X]$ has at most two roots in $A$, then…
TerranDrop
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(Dummit's AA, 1.5, P3) Are these presentations of the Quarternion group equivalent?

For example, from wiki, we know that $$ \langle i, j \mid i^4 =1, i^2 = j^2, j^{-1}ij = i^{-1} \rangle = Q $$ where $Q$ denotes Quaternion group. And by my own inspection, I speculated that $$ \langle i,j \mid i^4 = j^4 = 1, ij = j^3i \rangle =Q…
le4m
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