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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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Counting subgroups of a $p$-group.

Let $G$ be a finite $p$-group, say $|G|=p^n$, and let $0\le k\le n$. Call $\mathcal{A}$ the set of subgroups of order $p^k$, and $\mathcal{N}\subseteq\mathcal{A}$ the subgroups that are normal. I want to show that $|\mathcal{A}|\equiv…
Sandor
  • 565
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Finding the inverse of $2+\sqrt{5}+2\sqrt{7}$

Finding the inverse of $2+\sqrt{5}+2\sqrt{7}$ in the field $\mathbb{Q}(\sqrt{5},\sqrt{7})$. I know that all the elements of $\mathbb{Q}(\sqrt{5},\sqrt{7})$ are of the form: $a+b\sqrt{5}+c\sqrt{7}+d\sqrt{35}$, where $a,b,c,d \in \mathbb{Q} $ So I…
King Ghidorah
  • 607
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unknown permutation

Can someone help on this. I'm stuck on this part. I am trying to find a permutation $\sigma$ such that $$\sigma(1,2)(3,4)\sigma^{-1} = (5,6)(3,1)$$. By a particular theorem, I know I can have this one $$ \sigma (1,2)\sigma^{-1}\cdot…
marg_ocruz
  • 655
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Prove that the alternating group has a subgroup of order 12

Prove that $A_{5}$ has a subgroup of order $12$. What I have so far: The order of an alternating subgroup $A_{n}$ = $n!/2$. So the order of $A_{5}$ is $60$. We know that the order of a subgroup must be divide the order of the group, and this is the…
EmaLee
  • 1,153
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Index of a finite group

Prove that if G is a finite group, the index of Z(G) cannot be prime. What I have so far: -Suppose G is Abelian, then G = Z(G). In this case the order of G:Z(G) would just be 1 which isn't prime. What next?
EmaLee
  • 1,153
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Is $a$ is transcendental over Q if Q(a) = Q(x)? I know the converse is true.

I can prove the converse but not this direction. In fact, the converse is proved in Gallian. But I have yet to see a proof in this direction. Is this certainly true? Necessarily true? I am looking for a proof by contradiction. If a is algebraic…
OLP
  • 561
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Quotient groups of $\mathbb{RxR}$

Quotient groups of $\mathbb{RxR}$ In each of the following, H is a subset of $\mathbb{RxR}$. a) Prove that H is a normal subgroup of $\mathbb{RxR}$ (Remember that every subgroup of an abelian group is normal) b) In geometric terms, describe…
Emmie
  • 447
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example for torsion-free but not cyclic group

I am asked to provide an example for torsion-free but not a cyclic group. Is $\mathbb{Z}\times\mathbb{Z}$ an example?
breezeintopl
  • 1,437
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Show that c belongs to the commutator subgroup

Let a, b be elements in a group G and $c = a^{k_1}b^{l_1}a^{k_2}b^{k_2} \dots a^{k_n}b^{l_n}$ where $k_1 + k_2 + · · · + k_n = l_1 + l_2 + · · · + l_n = 0$. Show that $c$ belongs to the commutator subgroup $G´$, thus can be written as a product of…
Olba12
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The function $f:G \rightarrow G$ defined by $f(x)=x^2$ is a homomorphism iff G is abelian.

Let G be a group. The function $f:G \rightarrow G$ defined by $f(x)=x^2$ is a homomorphism iff G is abelian. I am having trouble with the 2nd part of the proof. Proof: Assume the function $f:G \rightarrow G$ is defined by $f(x)=x^2$ then: Assume…
Mark
  • 427
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Prove the following is a homomorphism and describe its kernel.

Prove the following is a homomorphism and describe its kernel. The function $f: \mathbb{RxR} \rightarrow \mathbb{R} $ given by $f(x,y)=x+y$ I just want someone to confirm my answer: My answer: Since $\mathbb{RxR}$ and $\mathbb{R}$ are both…
kero
  • 1,814
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Definition of "closure under isomorphism"

What does " Closed Under Isomorphism " means ? why do we need it ? And how can we use it in " Mathematical Structures" thanks
M.M
  • 256
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If $G$ has an element of order $p$ and an element of order $q$, where $p$ and $q$ are distinct primes, then the order of $G$ is a multiple of $pq$

If $G$ has an element of order $p$ and an element of order $q$, where $p$ and $q$ are distinct primes, then the order of $G$ is a multiple of $pq$ Here is how I am working out my proof: Suppose $x,y \in G$ and let $|x|=p$ and $|y|=q$ where $p$…
kero
  • 1,814
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Visualizing Third isomorphism theorem

That is the statement of the third isomorphism theorem. Let K and B be normal subgroups of a group G with N $\subset$ K $\subset$ G. Then K / N is normal subgroup of G / N, and the quotient group $(G / N) / (K / N)$ is isomorphic to G / K. I have…
user111750
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2 answers

Finding the kernel of a homomorphism mapping polynomials to polynomial functions

Consider the ring homomorphism $\phi: A[t]\to A^A$ where $A$ is an integral domain and $A^A$ is the ring of all functions on $A$ with values in $A$. Determine the kernel of $\phi$. Be as explicit as possible. I reasoned that if $f$ has $n$ distinct…
chris
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