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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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Conjugate subgroups are normal?

Suppose $A \subset C$ and $B \subset C$. Assume $A$ and $B$ are conjugate subgroups, that is $cAc^{-1}=B$ for some $c \in C$. Is the following statement true? $A=B$ if and only if $A$ and $B$ are normal subgroups of $C$ Question: If we assume $A$…
Idonknow
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Are there upper bounds for the number of groups of order $n$ as a function on $n$, for some special $n$?

If $n$ is prime there is only one isomorphism class of groups of order $n$, namely the cyclic group. I was wondering if there are other non-trivial examples of numbers $n$ where a non-trivial bound on the number of non-isomorphic groups of order $n$…
user2566092
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Is this question asking too much?

Question: Let $N\triangleleft G$ and $K\triangleleft G$. If $N\cap K=\langle e\rangle$ and $N\vee K=G$, then $G/N\cong K$. I used the second theorem of isomorphisms and this one (From Hungerford's book): Thus: $K\cong K/\langle e\rangle\cong…
user42912
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Every Ideal in R/K is of form I/K

A problem from Intro to Abstract Algebra from Hungerford. a) Let K be an ideal in a ring R. Prove that every ideal in the quotient ring R/K is of the form I/K for some ideal I in R. This is what I've done. Consider the function $f:R\rightarrow…
zzz2991
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a group of order $9$ is abelian

I want to show that a group of order $9$ is abelian without using "Sylow Theorem". I thought of using the theorem"If $$\frac{G}{Z(G)}$$ is cyclic then $G$ is abelian". Since $Z(G)$ is normal in $G$ , the factor group is well defined. Moreover since…
tattwamasi amrutam
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FInd gcd of two polynomials using Euclidean Algorithm

Let $f(x)=2x^4 +3x^3 −19x^2 −28x+ 6$ and $g(x)=x^3 + 2x^2 -9x -18$ be polynomials in $\mathbb Q[x]$. Use the Euclidean Algorithm to determine the gcd in $\mathbb Q[x]$. So far, I have the following: $2x^4 + 3x^3 -19x^2 -28x +6 = 2x(x^3 +2x^2 -9x…
Katherine
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Minimal $n$ for embedding of an abelian group into a permutation group $S_n$

Given a finite abelian group $G$, is there a formula or quick algorithm to determine the minimum $N$ so that $G$ can be embedded into the permutation group $S_N$? If $G$ is cyclic of order $n$ I believe I know the answer: Write $n$ as a product of…
user2566092
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Associative binary operation with an identity, and all elements have finite order. Show this is a group.

Let $S,\star$ be an associative binary structure with identity $e$. Assume that for every $s\in S$ there is an integer $n_s>0$ such that $s^{n_s}=e$. Show that $S,\star$ is a group. I am not sure what to do for this problem. Any feedback would be…
user139154
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A question on Euclidean Domains

Let $D$ be a Euclidean domain whose function δ satisfies: $$ (1) \qquad δ(ab)=δ(a)δ(b)$$ $$ (2)\qquad δ(a+b)≤ \max(δ(a),δ(b)). $$ Show that either $D$ is a field or $D=F[x]$, $F$ a field, $x$ an indeterminate. Thanks for reading!
gottigen
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$H$ a subgroup of $G$ implies a surjection $G\to H$?

Suppose $H$ is a subgroup of $G$. Is there always a surjective homomorphism $G\to H$? I am confused with this. Today in my algebra class we had a group $G$ of order $p(p-1)$ for $p$ prime and the teacher said that because of Sylow's theorem $G$…
Spenser
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What does "well defined up to isomorphism" mean?

My Algebra textbook says the following: $A\cup B$ is defined as the union of $A'$ and $B'$, where $A'$ and $B'$ are isomorphic to $A$ and $B$ respectively. Hence, $A\cup B$ is not well-defined as a set, but it is well defined up to…
algebraically_speaking
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Lowest product of pair multiplication

This is kind of an algebra question, and I am interested in an algebric proof to it. Suppose we have $k$ natural numbers that are all greater than $0$. We would like to arrange them in multiplication-pairs of two, such that the sum of each pair's…
TheNotMe
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What structure does this set of mapping have?

Suppose two sets $B = \mathbb R$, and $A$. $F$ is a set of mappings from$A$ to $B$, such that $\forall f_1, f_2 \in F$, there exists a bijection $g: B \to B$ , such that $f_2 = g(f_1)$, equivalently, $f_1 = g^{-1}(f_2)$. What algebraic structures…
Tim
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In general, is $U(R\times S) \cong U(R)\times U(S)$? How do I show it?

Some context: I'm trying to find $\text{Aut}(\mathbb{Z}_n)$ for any natural number $n$ (Problem I.2.15c in Hungerford, if you're wondering), and I'm using the Chinese Remainder Theorem to deduce that if $$n=p_1^{r_1}\dots p_k^{r_k},$$ then…
Reeve
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Proof showing that a group must be finite of prime order

I'm trying to prove the following: G is a group with order $\ge 2$ with no proper, non-trivial subgroups. G must be finite of prime order. My attempt: Consider $g \neq e \in G$ (we can do this since order of $G$ is at least 2). Since $G$ has no…
anon_swe
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