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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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coloring a tetrahedron (abstract algebra)

Consider a regular tetrahedron. Each of its faces can either be painted blue or red. Up to rotation how many ways can the tetrahedron be painted? I am thinking that applying Burnside's counting formula will work, but it has been so long since I have…
user113561
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$\exists N\leq M$ such that $K_{i}=N\cap M_{i}$ and $N_{i}=\pi_{i}\left(N\right)$($i=1,2$) ($\pi$ is projection)

Let $M$ be direct sum of two submodules $M_{1},M_{2}$. Prove that if $K_{i}\leq N_{i}\leq M_{i}$ ($...\leq...$ : $...$ be submodule of $...$) and $N_{1}/K_{1}\cong N_{2}/K_{2}$ then $\exists N\leq M$ such that $K_{i}=N\cap M_{i}$ and…
user111636
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if $N$ is submodule of $M$ then $\pi_{1}\left(N\right)/\left(N\cap M_{1}\right)\cong\pi_{2}\left(N\right)/\left(N\cap M_{2}\right)$

Let $M=M_{1}\oplus M_{2}$ ($M_{1},M_{2}$ are submodules of $M$). $\pi_{i}\left(i=1,2\right)$ is projection form $M$ to $M_{i}$. Prove that if $N$ is submodule of $M$ then $\pi_{1}\left(N\right)/\left(N\cap…
user109584
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Let $\left\{ f_{i}:M_{i}\longrightarrow N\right\} _{i\in I}$, can $Im\left(\oplus_{i\in I}f_{i}\right)=\sum_{i\in I}Im\left(f_{i}\right)$

Let $\left\{ f_{i}:M_{i}\longrightarrow N\right\} _{i\in I}$ be a family of $R$-homorphism from $R$-module $M_{i}$ to $R$-module $N$. Do we have $Im\left(\oplus_{i\in I}f_{i}\right)=\sum_{i\in I}Im\left(f_{i}\right)$?
user109584
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Bijective hom sets

Let $ f: G \to H $ be a group homomorphism. Suppose that the induced map $ F: \text{Hom}(H,H) \to \text{Hom}(G,H) $ defined by $ F(\phi) \stackrel{\text{def}}{=} \phi \circ f $ is a bijection. Show that if $ G $ is abelian, then so is $ H $. I'm…
Joe
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Characteristic of a field $F$ is prime

If Char$F$ $\neq 0$, then Char$F$ must be prime number. MY try: If Char$F$$ = nk $ for integers $n$ and $k$, then by definition, $nk = 0 \implies n = 0$ or $k = 0$ which implies Char$F=0$ which is a contradiction. Is this correct?
ILoveMath
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Prove or disprove: There exists a ring homomorphism $\phi: \mathbb{C}\rightarrow \mathbb{R}\times\mathbb{R}$.

Prove or disprove: There exists a ring homomorphism $\varphi: \mathbb{C}\rightarrow \mathbb{R}\times\mathbb{R}$. I think it is intuitive to try $\varphi: \mathbb{C}\rightarrow \mathbb{R}\times\mathbb{R}$ defined by $\varphi: a+bi\mapsto (a, b)$.…
LaTeXFan
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$HK$ is a subgroup where $H \subseteq N_G(K)$

Let $H, K$ be subgroups of $G$. Let $HK=\{hk: h \in H, k \in K\}$ and $H \subseteq N_G(K)$. I'm a bit stuck on something. It makes me a bit nervous that $H \subseteq N_G(K)$. Does this mean that $HK=KH$ or for every $h \in H$ and $k \in K$…
emka
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$\mathbb{Z}[i]/(a+bi) \cong \mathbb{Z}_{a^2+b^2}$ if $(a,b)=1$

I hope to show that $$\mathbb{Z}[i]/(a+bi) \cong \mathbb{Z}_{a^2+b^2}$$ for $(a,b)=1$. I made an effort to find a homomorphism but I failed. Can you give a hint?
Guillermo
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subgroups of order $p^2$

How many subgroups of order $p^2$ does the abelian group $\mathbb{Z}_{p^3} \oplus \mathbb{Z}_{p^2}$ have? Hint: Every cyclic group of order n contains a unique subgroup of order m for all divisors m of n. Proof:there are two types of groups of order…
Ruth Gutierrez
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If $r, s, t \in R$, then $r \gcd (s, t)$ is associate to $\gcd(rs, rt)$.

I seem to be stumped on this question. For the setting, let $R$ be an integral domain and let $r, s, t \in R$. The question asks Show that $r \gcd(s, t)$ is associate to $\gcd (rs, rt)$ To start, let $d$ be some $\gcd$ of $s$ and $t$, and let…
tylerc0816
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Conjugacy class of a group

A group G of order 12, with conjugacy class of order 4 has trivial center. My attempt: $|C(x)|=4 \implies |Z(x)|=3$. This implies that Z(x) is a cyclic subgroup of order 3. Thus $Z(x)= \{1,x,x^{-1}\}$. We know the center of the group $Z(G) \subset…
user104221
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Isomorphic Gaussian Integers

Can we show that the ring of Gaussian integers $$\mathbb{Z}[\sqrt{17}]:=\{a+b\sqrt{17}:a,b\in\mathbb{Z}\}$$ $$\mathbb{Z}[\sqrt{11}]:=\{a+b\sqrt{11}:a,b\in\mathbb{Z}\}$$ equipped with standard addition and multiplication are not isomorphic?
David Simmons
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Transcendence degree

If one has an extension of domains $D_1\subset D_2$ such that the extension of fields of fractions $K_{D_2}/K_{D_1}$ is transcendental of degree $r$ (with $K_{D_i}$ the field of fractions of $D_i$), is it always possible to find $r$ algebraically…
rfauffar
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Let V denote the Klein 4-group. Show that $\text{Aut} (V)$ is isomorphic to $S_3$

After a week in my Abstract Algebra class, the professor proposed this as a problem. I'm not entirely sure where to begin. $ V = \{ e, \tau, \tau_1, \tau_2 \}$, so I'm not sure exactly what is meant by $\text{Aut} (V)$. Is it simply saying that the…
Anthony Peter
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