Mathematics Stack Exchange
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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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Prove that if H is a subgroup of G, then H is a normal subgroup of G iff $\forall x, y \in G$, xy $\in$ H iff yx $\in$ H.

So the problem is as follows: Prove that if H is a subgroup of G, then H is a normal subgroup of G iff the following condition holds: $$\forall x,y \in G, xy \in H \iff yx \in H$$ It's all completed now: We are given that H is a subgroup of…
Katie
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how to prove $G_1$ and $G_2$ aren't isomorphic?

Assume $$G_1=\mathbb Z_5 \times \mathbb Z_{5^2}\times \mathbb Z_{5^3}\times \mathbb Z_{5^4} \times\ldots$$ $$G_2= \mathbb Z_{5^2}\times \mathbb Z_{5^3}\times \mathbb Z_{5^4} \times \ldots$$ How do I prove $G_1$ and $G_2$ aren't isomorphic? I asked …
M.H
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GCD computations in $\mathbb{Z}[i]$

Problem statement: Find a generator of the ideal $(85, 1+13i)$ in $\mathbb{Z}[i]$, i.e., a GCD for $85$ and $1 + 13i$ by the Euclidean Algorithm. Do the same for the ideal $(47-13i, 53+56i).$ Can you please outline the steps, then I can practice…
mary
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Finite Field Extension

Suppose $E/F$ is a field extension of degree $n$. Does it follow that $E = F(a_{1}, a_{2}, \ldots, a_{n})$ for some $a_{i} \in E$? I feel like this is true, but I'm getting confused with all the definitions.
Shayla
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Does the set of all strings of letters form a group?

This is related to a course I'm taking in computer science theory. Let $\sum$ be an alphabet. Then the set of all strings over $\sum$, denoted as $\sum^*$ has the operation of concatenation (adjoining two strings end to end). Clearly,…
user63411
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Basic Abstract Algebra question

I was helping out a high school student with factoring when a noticed that polynomials that factor in $\mathbb{Q[x]}$ also factor in $\mathbb{Z[x]}$. I was wondering if there is a formal argument to be made here from this observation. For example,…
Teacher OfLife
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Let $A,B$ be two subsets of a finite group $G$. If $|A|+|B|>|G|$, show that $G=AB$

Let $A,B$ be two subsets of a finite group $G$. If $|A|+|B|>|G|$, show that $G=AB$. My attempt is : Since $|A|+|B|>|G|$, there exists one common element in both sets $A$ and $B$, say $g$. Then since $G$ is a group, by closure, $g^2 \in G$, which…
Idonknow
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The kernel of a surjective homomorphism is finitely generated

Let $M$ be a finitely generated $R$-module and $f:M \rightarrow R^n$ a surjective homomorphism. By letting $e_1,\dots,e_n$ be the standard basis of $R^n$ and choosing $u_i \in M$ with $f(u_i) = e_i$ for $1 \leq i \leq n$, (i) Show that $M =…
user46220
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What algebraic structure is the set of natural numbers and addition?

What algebraic structure is the set of natural numbers and addition? I understand that $$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$$ and $\mathbb{Z}$ and $\mathbb{Q}$ are rings and $\mathbb{R}$ and…
qazwsx
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Is $\mathbb Q(\zeta_6)=\mathbb {Q}(\zeta_3)$?

I got myself confused over the following: We have $$\mathbb Q(\zeta_3)=\mathbb Q(\exp(2\pi i/3))=\mathbb Q\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right)=\mathbb Q\left(-\frac{1}{2}+\frac{i\sqrt 3}{2}\right)=\mathbb Q(i\sqrt 3),$$ but also…
Buh
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Isomorphism of Direct Product of Groups

I have $H_1,H_2,\dots, H_n$ groups with the property $H_i\cong G_i$, where $G_1,\dots,G_n$ are also groups. It should be somehow easily followed that $G_1\times \dots\times G_n\cong H_1\times \dots\times H_n$. I would define a function…
Alexander
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Proving this group is Abelian

Let $(G,.)$ be a group where there exists an element $g \in G$ such that for any $x \in G$ it is the case that $x^3 = gxg$. I've been stumped on this one. All I have found is that $e = e^3 = geg = g^2$. Does anyone have advice on some starting…
user528465
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How to find all morphisms from $(\mathbb{N}, \mid)$ to $(\mathbb{N}, \mid)$?

I just need a small hint, not the full answer. I know that, if $f$ is a morphism, $a \mid b \implies f(a) \mid f(b)$ $a \mid b$ and $a \mid c \implies a \mid b+c$, so $f(a) \mid f(b), f(a) \mid f(c), f(a) \mid f(b + c)$. Also, $f(a) \mid f(b) +…
Gabi Purcaru
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Middle Cancellation in Groups

For a, b, c, d, x elements of a group G. If ab = cd does that mean that axb = cxd? What if ab = cd only in this one instance, does the equality still hold?
Tom Jones
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An example of symmetric associative increasing function which cannot be represented as addition

Let $X$ be some connected subset of $\mathbb{R}$. Let $f: X^2\to X$ have following properties: $\forall x, y$: $f(x, y)=f(y,x)$ (Symmetry) $\forall x, y, z$: $f(x, y)>f(x,z)\iff y>z$ (Strictly increasing on any argument) $\forall x, y, z$: $f(x,…
Serge Seredenko
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