Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, and other advanced topics.

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Is there a generalization of the twin prime conjecture to rings or certain rings?

The question's in the title. For instance, if $R$ contains $2$ then there are an infinite number of pairs of prime principal ideals $(p),(q)$ such that $p = q + 2$. I just made that up and it's ...
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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 ...
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Tensoring and retaining projectiveness

Let $A$be a unital associative ring, If $A$ is not a projective $A$-bimodule, however $A\otimes A$ is a projective $A$-bimodule, can we conclude that $A\otimes A \otimes A$ is also projective?
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Subgroup of order $p$ is normal

I am trying to show that an arbitrary group $G$ of order $p^n$ has a normal subgroup of order $p$. My first instinct is to say that by Cauchy's Theorem, there is some element $x \in G$ such that the ...
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Matrix linear algebra generators

Linear algebra and special-linear group experts please help: It is known that in principle one can generate this $C$ matrix form the $A$ and $B$ matrix below. Here $$ C=\begin{pmatrix} 0& -1& ...
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Conjugacy classes

I don't seem to be able to follow this part of the proof. Why are we able to say $x \sim y$? Why does it suffice to prove that $y \in H$? Let $G$ be group. $(a)$ We say that elements $x,y \in ...
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81 views

“Abstract nonsense” proof of the splitting lemma

As remarked in the "talk" part of the wikipedia article, the proof is done with elements of a set and functions. I guess it's possible to carry it out purely with "objects" and "arrows" Who ...
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Boolean-like algebra

Suppose one had an algebra that that follows most of the laws of Boolean algebra (associative, commutative, distributive, identity, annihilator, idempotent, double negation, De Morgan) but does not ...
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Show that $(\mathbb{Q}^*,\cdot)$ and $(\mathbb{R}^*,\cdot)$ aren't cyclic

I'm reading a book about abstract algebra, but I'm having trouble solving this excercise: "Show that $(\mathbb{Q}^*,\cdot)$ and $(\mathbb{R}^*,\cdot)$ aren't cyclic" Where $(\mathbb{Q}^*,\cdot)$ is ...
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57 views

Completion of integral domain

Let $A$ be an integral domain with the $I$-adic filtration. Let $B$ be the fraction field of $A$. My question is the following: Is the fraction field of the completion of $A$ the same as the ...
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Problem of Galois Extension

$\Bbb K$ is a non-Galois extension of $\Bbb Q$ and $[K:\Bbb Q]=4$. If $\Bbb F$ is the Galois closure of $\Bbb K$ then show that $Gal(\Bbb F/\Bbb Q)$ is either $S_4, A_4$ or $D_8$ with order 8. ...
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Degree of non-separable extension

Suppose $K$ is a field with charasteristic number $p$. Further suppose that L is a non-separable extension of K with $[L:K]=k$. Why does it hold that k is a multiple of p?
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Can a ring isomorphism change the structure of a module?

Let $M$ be an $R$-module, where $R$ is a ring with unit. Given a ring automorphism $\phi: R \rightarrow R$, we can define a new $R$-module structure on $M$ by $r \cdot x = \phi(r) x$ for all $r \in ...
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Subgroups of direct products

Consider a group $G$ which is a direct product of two groups of coprime order: $G = G_1 \times G_2$ with $|G_1|=n_1$, $|G_2|=n_2$ and $\textrm{gcd}(n_1, n_2)=1$. Let $H \le G$. Is it true that ...
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60 views

When is $\langle x,y\rangle$ equal to $\langle x\rangle\langle xy\rangle$?

I would like to know necessary and sufficient conditions on $x$ and $y$ to have $\langle x,y\rangle=\langle x\rangle\langle xy\rangle$. Sure that: $\bullet$ $\langle x\rangle$ and $\langle ...
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Galois Theory problem (primitive roots of unity)

If $e_1,e_2......,e_{p-1}$ denote primitive $p^{th}$ roots of unity. Here $p$ is prime. And set the sum of the $n^{th}$ powers of the $e_i$ as $p_n=e^n_1+e^n_2......+e^n_{p-1}$ Now I want to show that ...
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Find all integers $m$ and positive integers $n > 1$ so that $m + \sum_{k=1}^n x^k/k!$ has a rational root

If $m = 1$, then $m + \sum_{k=1}^n x^k/k!$ has no rational root for $n > 1$. And clearly the polynomial has a rational foot for all integers $m$ if $n = 1$. So, besides those cases, for what ...
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Quarternion group

I'm having issues showing each element has an inverse. I could do it the arduous and check for all 8 elements way but I'd rather understand the solution in 3). I understand that the inverse is ...
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let $S_4$ denote the group of permutations of $\{1,2,3,4\}$ and let $H$ be a sub group of $S_4$ of order $6$ .

Let $S_4$ denote the group of permutations of $\{1,2,3,4\}$ and let $H$ be a sub group of $S_4$ of order $6$ . Show that $\exists~ i \in \{1,2,3,4\}$ which is fixed by each element of $H$. Attempt: ...
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Show that $G$ is a group if the cancellation law holds when identity element is not sure to be in $G$

Having already read through show-that-g-is-a-group-if-g-is-finite-the-operation-is-associative-and-cancel, however in Herstein's Abstract Algebra, I was required to prove it when we're not sure if ...
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20 views

Isomorphisms between groups

I'm not quite following the section of the solution highlighted. Why if (yx)^2 is equal to an element of order p, does that in turn mean yx has order p?
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Set of non-units in a ring

Let $R$ be a ring with identity. Let ${\rm rad}\: R$ be the radical of $R$, ie the intersection $\bigcap L$ over all maximal left ideals $L$ in $R$. Let $S$ be the set of all non-units in $R$ ...
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The field of algebraic numbers in $\mathbb Q (a_1,\ldots, a_l)$ is finite over $\mathbb Q$

In the book 'Algebra IV: Infinite Groups, Linear Groups' by Kostrikin and Shafarevich, there is a sketch of a proof (on page 84) of a theorem by Schur. I'm struggling to understand the line: Since ...
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3answers
64 views

The implication sign of Group Closure

I know that $x, y \in G$ implies that $xy\in G$, but does the implication go the other way as well?
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Describe the group automorphism of a k-automorphism

Let $B = k[T]$ with $k$ a field; a $k$-automorphism of $B$ is a ring homomorphism $φ: B \to B$ that is the identity on $k$ and is an automorphism of $B$; (i) Describe the group $Aut(B)$ of ...
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Let $K$ be a Sylow subgroup of a finite group $G$. Prove that if $x \in N(K)$ and the order of $x$ is a power of $p$, then $ x \in K$.

Let $K$ be a Sylow subgroup of a finite group $G$. Prove that if $x \in N(K)$ and the order of $x$ is a power of $p$, then $ x \in K$. This is how I tried... Since $K$ is normal in $N(K)$, the ...
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Find $b \in G$ for each $a \in G$, such that $b^m = a$, where $m$ is coprime to the order of group G

Let $(G, \cdot)$ be a finite group (with identity element $1 \in G$) and $m \in \mathbb{N}$ be coprime to $|G|$. Proofe that for all $a \in G$ there exists exactly one $b \in G$, such that $b^m = a$. ...
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Abstract Algebra: If $H$ and $K$ are subgroups of an abelian group $G$ of order $m$ and $n$, prove $G$ has a subgroup of order $\mathrm{lcm}(m,n)$

Let $G$ be an abelian group and $H$ a subgroup of order $m$ and $K$ a subgroup of order $n$. Prove that G has a subgroup of order $\mathrm{lcm} (m,n)$ (lcm = least common multiple). any thoughts? ...
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How to show that complement of prime filter is ideal? [closed]

How to show that in any lattice L, F is a prime filter if an only if its complement L\F is an ideal?
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For a shift matrix $A$, prove that $A^n=0$ but $A^{n-1} \neq 0$.

Let $A\in F_n$ be the matrix $\begin{pmatrix} 0&1&0&0&\cdots&0 \\ 0&0&1&0&\cdots&0 \\ \vdots\\ 0&0&0&0&\cdots&0 \end{pmatrix}$, whose ...
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Show that A(T) is a group under the operation of composition of functions.

Let T be the non-empty set and A(T) the set of all permutations of T. Show that A(T) is a group under the operation of composition of functions.
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Prove a nonzero ring is not a group.

Prove that a nonzero ring R is not a group under multiplication. [Hint: what is the inverse of 0?] I know that if R is a nonzero ring the for x,y in R xy=0 means either x=0 or y=0 and when x is not = ...
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Order of Element in Factor Group

Give the order of the element in the factor group: $$26 + <12> \text{ in }Z_(60) /<12>.\; (26 + <12>) = (2 + <12>) = {2, 14, 26, 38, 52} $$ yeah? This has order 5 so I ...
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why are these cosets equal?

Please disregard this question until I have uploaded a screenshot K is the subgroup of S_3 defined by the permutations {(1), (123), (132)} They have (1)K = (12)K = {(1), (12)} What they did was ...
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double coset represntatives

Let $H$ be a subgroup of finite index in the group $G$. Let $g\in G$. We use $r\in HgH/H$ as notation for $g\in R$, where $R$ is a complete set of representatives for $HgH/H.$ Proof or disproof: If ...
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Prove that the well ordering principle is equivalent with PMI.

So I am supposed to prove that the well ordering principle is equivalent with the maximum principle. Well ordering principle: Every nonempty subset of the set of positive integers has a least ...
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Check if $\phi : \mathbb{Z}_9 \to \mathbb{Z}_2$ is a Homomorphism

Check if $\phi : \mathbb{Z}_9 \to \mathbb{Z}_2$ is a Homomorphism I say, let $x,y \; \in \mathbb{Z}_9$. $$\phi(x+y) = \phi(x+y)\pmod{2} = x +_2 y = \phi(x) + \phi(y)$$ However, this is wrong and I ...
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Identifying the Galois Group $G(\mathbb{Q}[\sqrt{2}, \sqrt[3]{2}]/\mathbb{Q})$

I am trying to determine the Galois group $G(\mathbb{Q}[\sqrt{2}, \sqrt[3]{2}]/\mathbb{Q})$. I am fairly confident I have the correct answer, but I need someone to confirm my work since I have just ...
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Nilpotent group and center

Let $G$ a nilpotent group. Show that (i) If $N$ is normal to $G$ and $N \neq 1$, then $N \cap Z(G) \neq 1$ (ii) If $N$ is normal to $G$ with $|N|=p$, where $p$ is a prime number, then $N \le Z(G)$. ...
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Coordinate ring of the unit circle is never a UFD?

I'm reading some notes about coordinate rings. On the third example on the second page, the author notes that the coordinate ring $K[\mathcal{C}]$ is not a UFD. If $f=X^2+Y^2-1$, then in ...
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Coset,sylow theorems

If $|G|=n$ and $p^\alpha | n$ then $ \exists H\leq G$ with $|H|=p^\alpha$. I got stuck in one part of the proof, can anyone clarify? (induction on $n$) Suppose for inductive hypothesis that ^ holds ...
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Locally nilpotent elements in modules

While reading through Lang's algebra, I came across following definition and proposition. In my opinion, something seems wrong. Some lemmas Lemma 1. Let $S$ be a multiplicative subset of $A$, and ...
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Free $A$-module isomorphic to a direct sum of copies of $A$?

Does this proposition hold even if it's not finitely generated? I think it does, since $M$ isomorphic to the direct sum of $M_i$, $M_i$ isomorphic to the direct sum of ...
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Is the neutral element unique in non-abelian groups?

I found different definitions of a 'group'. One stated that the neutral element has to be unique, the other only that it had to exist. I assumed that this means that the neutral element will be unique ...
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Show that $(x-a,x-b)=1$

Knowing that $K$ is a field, $a,b \in K$ different from each other,show that $x-a,x-b$ co-primes. We suppose that $\exists f(x) \in K(x)$ such that: $f(x)|x-a$ and $f(x)|x-b$ Then $\deg f(x) \leq ...
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Definition of field extension

One usually defines field extension $E/F$ whenever $F\subseteq E$. However, few authors would define field extension $F/K$ whenever there is a nonzero field homomorphism $F\rightarrow E$ (see e.g ...
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Noetherian local ring, detail in theorem 1.3.16 in Liu

I can't understand a detail in the proof of theorem 1.3.16 in Liu. The theorem is: let $(A,\mathfrak{m})$ a Noetherian local ring, $\hat{A}$ its $\mathfrak{m}$-adic completion, $(B,\mathfrak{n})$ an ...
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Automorphism Groups of Lie Groups

Take $X$ to be a Lie group and $Aut(X)$ to be its automorphism group (group isomorphisms which are also homeomorhisms). In general, are there some Lie groups in which this can be computed? For ...
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Prove that, up to isomorphism, there are only finitely many groups of size n.

I thought i understood the term "up to isomorphism" but in this case it seems not. Surely if your group elements are fixed there is only one possible group?
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Is Modeling the Future of Mathematics [closed]

So my Differential Equations professor today said one sentence then left the room. It was the weirdest moment of my college career but I felt like he really tried to say something. He walked in 1 ...