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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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Why is it that for $n<60$, the simple groups are precisely the cyclic groups $\mathbb{Z}_N$ for prime $n$?

I was reading this Wikipedia article and came across this For groups of order $n<60$, the simple groups are precisely the cyclic groups $\mathbb{Z}_n$ for prime $n$. I was wondering, why is this true for $n<60$? What is so special about this…
Trogdor
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What groups can I safely refer to, to demonstrate various theorems in a first course in abstract algebra?

When studying abstract algebra, I do prefer having a nice simple and concrete example to demonstrate the theorem/lemma. However, the 'first course' book that I am currently learning from often uses (in my opinion) quite complicated examples to…
Trogdor
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Example of nonAbelian group with a normal subgroup such that the quotient group is Abelian?

Can someone give me an example of a non-Abelian group $G$ with a normal subgroup $H$ such that $G/H$ is Abelian?
Fred
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Show that $\sigma\circ\phi : G_1 \to G_3$ is an isomorphism

Let $G_1, G_2$ and $G_3$ be groups. Let $\phi: G_1 \to G_2$ and $\sigma: G_2 \to G_3$ be isomorphisms of groups. Show that $$\sigma\circ\phi: G_1 \to G_3$$ is an isomorphism. I understand to prove the composition is a homomorphism (operation…
maidel b
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About Sylow's 2nd and 3rd theorems

I just do not understand one step of proof of Sylow's second theorem given in my textbook Basic Abstract Algebra by P. B Bhattacharya. The step is, as K is a Sylow p-subgroup of G, hence $gcd(|G/N(K)|, p) = 1$. How it comes? Here, the notation…
Akash Patalwanshi
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Elegant way to show that N is a normal subgroup of G

Claim: Let $G$ be the set of all real $2 \times 2$ matrices $\left( \begin{array}{cc} a & b \\ 0 & d \end{array} \right)$ such that $ad \not = 0$, with matrix multiplication as the operation. Let $N$ be the subset where $a = d = 1$. Then $N$ is a…
dalastboss
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Isomorphism classes of $\mathbb{Z}[i]$ modules.

$\textbf{Question:}$ How many isomorphism classes of $\mathbb{Z}[i]$-modules with exactly $5$ elements are there? $\textbf{My Attempt:}$ Since $\mathbb{Z}[i]$ is a P.I.D and any module with $5$ elements is finitely generated we can use the structure…
user7090
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If $p(x,y) \in \mathbb{Q}[x,y]$ is such that $p(x,e^x) = 0$, then $p(x,y) = 0$ too.

Suppose that for some two variable rational polynomial that evaluation of the second variable at $e^x$ gives 0, then the evaluation at any $y$ also gives 0. I've seen a proof that uses a calculus-based limiting argument, but I was wondering if there…
walkar
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Suppose $F$ is a field and the irreducible polynomial over $F$ of $x$ is of odd degree.

A question I recently encountered is: Let $F$ be a field. Let $x$ be algebraic over $F$. Suppose the minimal polynomial of $x$ is of odd degree. Show $F(x) = F(x^2)$. We know $F(x^2)$ is a subset of $F(x)$ because $x^2$ belongs to $F(x)$. To show…
ThinkConnect
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Classify groups of order $2014$

This is a question from an old exam. There should be $4$ groups of order $2014.$ Note $2014 = 2 · 19 · 53$. Admittedly there is an answer Can only find 2 of the 4 groups of order 2014? but I can't make much sense out of the attempt nor the…
cap
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Showing that $7+\sqrt[3]{2}$ is an algebraic number

How do I go about showing that $7+\sqrt[3]{2}$ is an algebraic number? I need to show that it is the root of an integer valued formal polynomial? How do I solve these problems in general? I haven't a clue where to start on this one, so it is hard to…
Oceans Bleed
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Show that If $R[X]$ is Euclidean domain then $R$ is a field

Let $R$ is an integral domain . Show that If $R[X]$ is Euclidean domain then $R$ is a field . I'll be waiting for your help. Thank you very much in advance!
Sara
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Alternating Group $A_n$ does not have proper subgroup of index less than n, where n>4.

A proof is to be given for this. So what i have thought is: Let us assume to the contrary, i.e. it does have a subgroup of index m (say) less than n. Then, since $A_n$ is simple for n>4 , by embedding theorem, $A_n$ is isomorphic to a subgroup of…
Jasmine
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Definition of $\mathbb{Z}[\omega]$ where $\omega$ is a primitive root of unity

What does $\mathbb{Z}[\omega]$ usually mean when $\omega$ is a primitive root of unity?
Mykie
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Number of orbits for the action of Galois group G.

Let G be the Galois group of a field with nine elements over its subfield with three elements. Then find the number of orbits for the action of G on the field with 9 elements.
Surya Deep Mishra
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