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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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Is pair $(m, -m)$ which $m\in \mathbb{Z}$ is isomorphic to $\mathbb{Z}$?

Is $\{(m, -m)\mid m\in \mathbb{Z}\}$ is isomorphic to $\mathbb{Z}$? What i know about isomorphic is, simply it is a bijective homomorphism. How one can define isomorphism map $\phi$ in above case?
phy_math
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If $G$ is non-Abelian, show that $Aut(G)$ is not cyclic.

If $G$ is non-Abelian, show that $Aut(G)$ is not cyclic. I can see that $G$ non-Abelian $\Rightarrow$ $G / Z(G)$ is not cyclic. And $G / Z(G)$ is isomorphic to $Inn(G) \Rightarrow$ $Inn(G)$ is not cyclic. But from here I can't see how I'd relate…
Oliver G
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Need help with fundamental theorem of homomorphisms

Theorem (FTH) Let $f: G \rightarrow H$ be a group homomorphism and let $N$ be a normal subgroup of G which is contained in $\ker(f)$ Then there exists a unique group homomorphism $\overline{f}:G/N \rightarrow H$ such that $\overline{f}\circ v=f$.…
daniel
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Question about the center of a group

Show that $Z(G)$ is subgroup of G. $Z(G)$ is the center of the group Proof: First we show $Z(G)$ is a group being a subgroup if $Z(G)$ is a group then 1) $1_G \in Z(G)$ 2) $Z(G)$ is closed 3) if $a \in$ then $a^{-1}\in Z(G)$ 1) $1_G*x=1_Gx$…
daniel
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When is an algebra map on generators well-defined?

Let $A$ be an $R$-algebra ($R$ comm. ring) and let $a,b$ a generating set of $A$, i.e. any element of $A$ can be written as products, sums, $R$-multiples of $a,b$. Then one can define any map $f:A\to B$ by its image on the generators. This is quite…
user360376
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If $a_1, a_2, ..., a_m$ each divide $n$, prove that $lcm(a_1,a_2, ..., a_m) | n$.

If $a_1, a_2, ..., a_m$ each divide $n$, prove that $lcm(a_1,a_2, ..., a_m) | n$. I see that essentially says show that the lcm of factors of $n$ divide $n$ and this makes sense intuitively, but I can't seem to show it. I can see that if…
Oliver G
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Is $S_3 \oplus \Bbb Z_2$ isomorphic to $A_4$ or to $D_6$?

I know $S_3 \oplus \Bbb Z_2$ is isomorphic either to $A_4$ or to $D_6$, where $S_3$ is the symmetric group of degree $3$, $A_4$ is the alternating group of degree $4$, $D_6$ is the dihedral group of order $12$, and $\oplus$ is the external direct…
Oliver G
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Example of a cluster algebra of geometric type and ground field

In most definitions of a cluster algebra of geometric type, it is said to be the subalgebra of the field $\mathbb Q (x_1,\cdots,x_n)$ generated by the cluster variables. I would like to compute such an algebra but I am not sure of the result. As an…
Bebop
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semisimple algebra

I just posted this on overflow.... just can't figure it out. Is there a direct proof of the following without going through composition series or Artin-Wedderburn theorem? Let V be a finite-dimensional complex Hilbert space. Let A⊂End(V) be a…
magya
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Let $p$ be prime. If an INFINITE group has more than $p-1$ elements of order $p$, why can't the group be cyclic?

Let $p$ be prime. If a group has more than $p-1$ elements of order $p$, why can't the group be cyclic? I understand how to prove this if the group is finite because the contrapositive of this statement is true due to the Euler-$\phi$ function $\phi…
Oliver G
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Let $|G|$ be an abelian group and let $H = \{g \in G : |g| \text{ divides } 12 \}.$ Prove that $H$ is a subgroup of $G$.

Let $|G|$ be an abelian group and let $H = \{g \in G : |g| \text{ divides } 12 \}.$ Prove that $H$ is a subgroup of $G$. I know that I have to show that $a,b \in H \Rightarrow ab^{-1} \in H$ or $(ab \in H \land a^{-1} \in H).$ But I can't figure…
Oliver G
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Generating a subfield with identity element

I know this is a very basic question but I am always unsure what exactly is meant by "generating". For example, consider the polynomial ring $k[x]$, and the ideal generated by $f(x)$, denoted by $\langle f(x) \rangle = k[x] \cdot…
GhTU
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reasoning about almost-groups and almost-associativity?

I have a group-like structure with a unique property that makes it not quite a group. I've been thinking of it as "almost associativity" and I'm wondering what's been studied about it? To see what I mean, first consider an operation $*$ that is…
Mike Izbicki
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$[L:K]\leq [K_1:K][K_2:K]$

Let $K_1,K_2$ be intermediate fields of field extension $K\subseteq L$ such that $L=K(K_1,K_2)$ ($L$ is the smallest field containing $K_1$ and $K_2$). I would like to prove that $[L:K]\leq [K_1:K][K_2:K]$. My idea is by starting with a basis…
user9077
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Characteristic of a field not equal to $2$

Let $E$ be a field with characteristic not equal to $2$. Let $F$ be an extension field of degree $2$. Prove that $F=E(\alpha)$ with $Irr(\alpha,E)=x^2-b$. Show that this is false for some field $E$ of characteristic $2$. My thought is let $p(x) =…
jacky123456789
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