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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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Simple Group isomorphic to $\mathbb{Z}_p$

Let $G$ be a group, $H$ and $K$ normal subgroups of $G$ such that $H$ and $K$ are simple, $G=HK$, and $H\cap K = \langle e \rangle$. Show that either $H\cong K\cong \mathbb{Z}_p$ for $p$ a prime, or The only normal subgroups of $G$ are $\langle…
Lila
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Prove that $G$ is either the trivial group $\{1\}$ or isomorphic to $\mathbb Z / 2 \mathbb Z$.

Problem: Suppose $G$ is a finite abelian group and there is no non-trivial isomorphism $f: G \rightarrow G$ which has order $2$. Prove that $G$ is either the trivial group $\{1\}$ or isomorphic to $\mathbb Z / 2 \mathbb Z$. My idea: Let $f: G…
Thu Le
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Is U-group isomorphic to $\mathbb{Z}_4 \oplus\mathbb{Z}_4$?

How can i show that no $U(n)$ is isomorphic to $\mathbb{Z}_4 \oplus\mathbb{Z}_4$? Please give me some hint. Thanks in advance.
Siddhant Trivedi
  • 2,298
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An equation in tensor product

I want to find the numbers of solutions of below equation:$$| \mathbb Z_n \otimes \mathbb Z_{12}| =\frac{n}{2} $$
rese
  • 1,054
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Is there a description of the ring of endomorphisms of $\mathbb{Q}$ as an abelian group?

Regarding $\mathbb{Q}$ as a field, there is only the trivial automorphism. What happens if we only consider $(\mathbb{Q},+,0)$ as an abelian group? Is there a description or way to compute the ring of endomorphisms $\mathrm{End}(\mathbb{Q},+,0)$?
madison flom
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Monoids. Disprove that $(a\cdot x=a) \Rightarrow (x=e)$

Wikipedia states that "...not every monoid sits inside a group. For instance, it is perfectly possible to have a monoid in which two elements a and b exist such that $a\cdot b = a$ holds even though $b$ is not the identity element." (link:…
Ralph
  • 331
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What proof techniques show that this polynomial is irreducible?

Here's my homework problem: Let $K = \mathbb{Z}_2[x] / \langle x^4 + x^3 + 1\rangle$. Show the polynomial $p(x) = x^3 + x + 1$ is irreducible in $K[x]$. Since the polynomial is third-degree, I suppose it would be sufficient to show that it has no…
GMB
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$2 \times 2 $ Orthogonal Group

I am trying to prove that $O(2)$ has two connected components. This is what I have done: Suppose $A \in O(2)$. Then $A^tA=I$, where $A^t$ is the transpose of $A$ and $I$ is the identity. Taking the determinant of both sides of this equation, we get…
Bob Joe
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Surjective map from polynomial ring over a field to the field.

Let $z\in R$ be fixed then the map $\phi:R[x]\rightarrow R $ defined as $\phi(f(x))=f(z)$ is surjective? Could someone please explain me why it is.
mmm
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group with property of order of operations unimportant being abelian?

Let $G$ be a group with the following property $$\forall a, b, c \in G,\quad ab = ca \implies b = c.$$ Show that $G$ is abelian. I know this hints towards the elements of the set being commutative but not sure. I understand that this property hints…
tomatoketchup
  • 59
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Three fields $F\leq F_1,~F_2$ and $F_1\neq F_2$ but $F_1\cong F_2$

I'm thinking about something in field theory. I extract the essence problem to ask here (so the problem may be weird). Given three fields $F,~F_1,~F_2$ with $F\leq F_1$, $F\leq F_2$, $F_1\neq F_2$ and $F_1\cong F_2$. Can it be shown that $F_1$…
Eric
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''Homomorphism'' of rings

Let $R$ and $R'$ rings, $\phi$ a homomorphism of $R$ to $R'$ such that $\phi(x+y)=\phi(x)+\phi(y), \quad \forall x,y \in R$. $\phi(xy)=\phi(x)\phi(y) \quad or \quad \phi(xy)=\phi(y)\phi(x) \quad \forall x,y \in R.$ Then $\phi(xy)=\phi(x)\phi(y)…
Juan Carlos Arroyave
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Does transitivity of a relation imply associativity of an operation?

Let $*$ be a binary operation on a set $S$. We define a binary relation $R$ on $S$ by: $xRy$ iff $\exists z: x*z=y$. If $*$ is associative, then the relation $R$ is transitive. My question is whether the converse is true.
user107952
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calculate the ideal generated by $\langle2,x\rangle$

I don't get it how calculate $\langle2,x\rangle$. I need that because I want to use that these ideals cannot be generated by one single element. And concludes that $\mathbb{Z}[x]$ is not a PID.
José Marín
  • 941
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4 answers

Number of elements in a factor ring

I am presented with $f(x) = 2x^3 + 3x^2 + 1$ $\in \mathbb{Z_5}[x]$ and need to explain why $F = \frac{\mathbb{Z_5}[x]}{f(x)}$ is a field and also find how many elements are in F. So far I have shown that $f(x)$ is an irreducible polynomial and I…
MathsRookie
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