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

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Automorphism of the subgroup [duplicate]

Let $G$ a finite group and $H<G$. Let $\varphi:H\rightarrow H$ be a nontrivial automorphism of $H$. My question is: it is possible to construct a nontrivial automorphism of $G$? I had tried to ...
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1answer
22 views

Every subgroup of finite index contained in an infinite group $G$ contains a normal subgroup of $G$. [duplicate]

Let $H<G$ such that $H$ has finite index in the infinite group $G$. Then $G$ contains a normal subgroup of finite index contained in $H$. Can I create a subgroup of index $2$ in $G$ using elements ...
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1answer
27 views

Submodules finitely generated if sum and intersection are finitely generated? [duplicate]

Let $M,N$ be submodules of an R-module $L$, where $R$ is a commutative ring with unity. I would like to prove that if $M+N$ and $M \cap N$ are finitely generated so are $M$ and $N$. Trying to combine ...
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2answers
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Finding the orbits of the orthogonal group $O(n)$ on $\Bbb R^n$

Let $O(n)=\{M\in GL_n(\mathbb{R}):MM^t=M^tM=I\}$ an orthogonal group. I need please an explain why each orbits consists of all vectors with the same length. I know that an orbit is defined by ...
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On describing a sort of “well-behaved” subgroups of a free abelian group.

I found this question when I tried to figure out what kind of subgroups of a free abelian group behave just as well as in the finite generated case. Let $M$ be an free abelian group, $N$ a subgroup ...
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1answer
34 views

Finding exact isomorphism between finite fields given as quotient rings [duplicate]

I have two quotient rings over $\Bbb F := GF(3)$: $$\Bbb F[x] / (x^3 -x - 1) \qquad \text{and} \qquad \Bbb F[x] / (x^3 -x + 1) .$$ These things I know: Both quotient rings are irreducible, that means ...
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1answer
20 views

Image and Kernel of composition of two homomorphisms

I have just showed that the composition of $a * b$ of two homomorphisms $a,b$ is a homomorphism. However, what can I say about the image and kernel of $a*b$, in terms of $a$ and $b$? Is there ...
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1answer
17 views

Is a ring R, modulo an ideal I (generated by x), then modulo an ideal J (generated by n) the same as R modulo the ideal generated by (n,x)?

Is the following statement true? $$ R/(x,n) = \left[ R/(x) \right] / (n) $$ My thinking behind it was as follows: \begin{array}{ccc} \left[ R/(x) \right] / (n) & = & \{ r+(n) : r \in R/(x) ...
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1answer
97 views
+150

Surjective exponentials for algebraically closed fields

The existence of the exponential on $\mathbb{C}$ has a very basic, yet very strong consequence : $(\mathbb{C}^*,\cdot)$ is a quotient of $(\mathbb{C},+)$. This question is concerned with fields $K$ ...
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+50

Cosemisimple Hopf algebra and Krull-Schmidt

A cosemisimple Hopf algebra is one which is the sum of its cosimple sub-cobalgebras. Is it clear that a comodule of a cosemisimple Hopf algebra always decomposes into irreducible parts? Moreover, will ...
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1answer
51 views

$|G| = 12p$ with $p>5$ is not simple

A group of order $12p$, $p>5$, is not simple, where $p$ is prime. [Hint: Consider three cases : $p=7$, $p=11$ and $p>11$.] My attempt : Let $G$ be a group. $|G| =12p$. Note that ...
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2answers
26 views

For $a_{i}:=\sqrt{2+a_{i-1}}$, how to prove that $\mathbb{Q}\subset\mathbb{Q}(a_i)$ is cyclic of degree $2^i$?

Let us define numbers $a_i\in\mathbb{R}_{\geq0}$ by $a_0=0$ and $a_{i+1}=\sqrt{2+a_i}$. How do I prove that $\mathbb{Q}\subset\mathbb{Q}(a_i)$ is cyclic of degree $2^i$? How do I prove that ...
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3answers
74 views

Showing $\mathbb{Q} \times \mathbb{Q}$ is not a field

I am revising and have come across the question Show that $\mathbb{Q} \times \mathbb{Q}$ with element-wise addition and multiplication is not a field I don't understand how to go about this, do i ...
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11 views

discrete valuation ring for formal series

as I can define a discrete valuation ring for formal series. I can define it as: Formal power series ring, norm.
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Any characterization of rings over which submodules of left free modules are free?

Let $R$ be a ring (with identity, but not necessarily commutative). If given the presumption that for any left free $R$-module $M$, every submodule of $M$ is free, what can we say about $R$? ...
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1answer
59 views

Extending a given element of a free abelian group to a basis

Let $V$ be a free $\mathbb{Z}$-module (or a free abelian group) with basis $(v_1$, $v_2$, $v_3)$. Denote by $x$ an arbitrary nonzero element of $V$, say $x=mv_1+nv_2+kv_3 \in V \setminus \{0\}$; here ...
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2answers
36 views

Finding counterexample in fields [duplicate]

Task is real simple to understand: I need an example of two fields, E and F, such that their additive parts are isomorphic, and their multiplicative parts are also isomorphic, but fields themselves ...
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0answers
27 views

finite groups all of whose maximal subgroups are cyclic

Let $G$ be a finite group and let all of maximal subgroups of $G$ are cyclic. What can we say about the structure of $G$?
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1answer
50 views

The ideals $\langle y-x-1\rangle$ and $\langle x-2,y-3\rangle$ in $\mathbb C[x,y]$ are prime

The following is a quote from Wolfram MathWorld article about prime ideals. A maximal ideal is always a prime ideal, but some prime ideals are not maximal. In the integers, $\{0\}$ is a prime ...
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Group Abstract-algebra

Recall that if $G$ is a group and $a,b ∈ G$ then $[a,b] =: a^{−1}b^{−1}ab$ is the commutator of $a$ and $b$ in $G$. Recall also that $[G,G] := \{[a,b]|a,b ∈ G\} ≤ G $ is the commutator subgroup of ...
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Why does dual basis not span the dual space when the given vector space V is infinite dimensional?

We have a vector space V with basis $\mathcal{A}$. I have to show that the set $\mathcal{A}$* = { v* | v$\in\mathcal{A}$} does not span the dual space V*. I can not see why dual basis fails to span ...
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1answer
37 views

Dimension about space of matrices of order 3 over the field of the real.

Consider the vector space of the matrices of order 3 over the field of the real $M_{3}\left(\Re\right)$ numbers. and let S be the subspace such that is spanned by the matrices of the form $AB-BA$. ...
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1answer
42 views

describe explicitly all the ideals of $R/(f(x))$

Let $R := \mathbb R[x]$ be the polynomial ring over the real numbers and $f(x) = x^3 - x^2 \in R$. Describe explicitly all the ideals of $R/(f(x))$ where $(f(x))$ is the ideal of $R$ generated by ...
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2answers
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Fibers of extension of scalars to algebraic closure of an affine variety is of dimension 0.

Let $X$ be an affine variety over $k$ and $\overline{k}$ be the algebraic closure of $k$. Then the projection morphism $X_\overline{k} = X \times _k \overline{k} \to X$ has dimension 0 fibers. ...
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0answers
50 views

Abstract algebra, Linear algebra 2, or Introduction to Topology? [closed]

Out of these which would you recommend taking over the summer? I'm kind of up in the air about it. Also do you know of any online resources for the following classes?
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0answers
112 views

Is $\dfrac{\cos\theta}{\sqrt{15}}$ irrational? [closed]

In general I was wondering if $\cos\theta$ was between $0$ and $1$ exclusive then would $\dfrac{\cos\theta}{\sqrt{15}}$ be irrational? And just on another note is an irrational times a transcendental ...
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0answers
10 views

LCM and the order of an element in a group [duplicate]

Tricky Question I am trying to solve. Let $o(a) = m$ and $o(b) = n$ in a group $G$. If $ab = ba$, show that an element $c \in G$ must exist such that $o(c) = lcm(m,n)$. There is a hint referenced to ...
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Epimorphisms and faithful functors in a rigid abelian tensor category

Let $\mathsf{C}$ be a rigid abelian tensor category in the sense of Deligne and Milne's notes (p.9). Let $\mathbf{1}$ denote the identity object in $\mathsf{C}$ with respect to $\otimes$. The abelian ...
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Difference between conjugacy classes and subgroups?

I am studying Group theory and Im not sure I understand the difference between a conjugacy class and subgroup?
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1answer
33 views

Let D be a Euclidean domain and d be the associated function.Show that if a and b are associates in D then d(a)=d(b). [closed]

I'm not even getting how to begin this.Any hints are welcome. Is < a >=< b > implies d(a)=d(b)? I have proved that if a and b are associates then < a >=< b >,but i'm not getting how to ...
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58 views

Notation in commutative algebra

I am doing some exercises on commutative algebra and came along the following expressions, which were not elaborated on. Is someone familiar with them? The first is for $p$ a prime number ...
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1answer
57 views

How many roots does the polynomial $p(z) = z^8 + 3z^7 + 6z^2 + 1$ have inside the annulus $1 < |z| < 2$?

How many roots does the polynomial $p(z) = z^8 + 3z^7 + 6z^2 + 1$ have inside the annulus $1 < |z| < 2$? I know I can use Rouche's Theorem. I'm just not sure how. It states that $|f(z) − g(z)| ...
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Primitive element in multivariate Galois field [closed]

On Singular CAS I can define a Galois field $(2^3)$ with $(x,y,z)$ variables. But I am not able to understand how $a^3+a+1$ is still its primitive element. General example taken in books is always ...
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1answer
36 views

Torsion-free divisible module over a commutative integral domain is injective. [duplicate]

This question is from the book Basic Algebra by P.M. Cohn. Show that a torsion free divisible module over a commutative integral domain is injective.
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Maximal Ideal of ring $C[0,1]$ [duplicate]

Prove that an ideal $M$ of the ring $C[0,1]$ is maximal iff there exists some $a$ in $[0,1]$ such that $M=\{f \in C[0,1]:f(a)=0\}$.
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2answers
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Why are there no “continuous maps” in algebra. [closed]

Or maps that behave similarly? Sorry if this is a strange question.
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1answer
48 views

Group generated by $x , y$ is non-commutative when $x^2 \cdot y^{-3} = I$.

The problem: Suppose group $G$ with generators $x$ and $y$ is defined by the relation $x^2 \cdot y^{-3} = I$. It is necessary to show that the group is non-commutative. I failed to solve the ...
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1answer
23 views

Sum of nil right ideals as an ideal

I have two questions: 1) If $S$ is the sum of all right nil ideals of a ring $R$ (with unity), is it true that $S$ is a two-sided ideal? It is clear for me that $S$ is a right ideal (and it is nil if ...
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1answer
52 views

Is it possible to say that if a A is a set of elements of a semigroup S, there exists some other semigroups on the alphabet A?

Suppose we have a finite semigroup, its element's set is A = {1,2,...n}, can we get some other semigroups whose element's set is A too ?
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The product of dg Lie algebras

I am trying to understand what are products and coproducts in the category of dg Lie algebras. I am okay with coproducts. For products, however, this Wikipedia article says that given ...
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Orthogonal projection of skew-symmetric form

It is a question from the book Algebra by Michael Artin: 8.8.2 Let W be a subspace on which a real skew-symmetric form is nondegenerate. Find a formula for the orthogonal projection $\pi:V\to W$
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Any nonabelian group of order 12 is isomorphic to A4, D6, or Z3 X Z4 [closed]

Can someone show me the proof for : Any nonabelian group of order 12 is isomorphic to $D_6$, $A_4$, or $\mathbb{Z}_3 \times \mathbb{Z}_4$ Ive seen a few proofs where this is included in also ...
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1answer
31 views

Show this product of principal ideals is principal

Let $R$ be a commutative ring with $1$ and $I$ an ideal. Also let $B$ be a principal ideal, and $A=\{a\in R\;|\; aB\subseteq I\}$. I want to show that if $A$ is also principal then $I$ is principal. ...
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A finite cancellative monoid is a group? Need help seeing why the following is a false proof.

I proved this in the following way is our exam, but got $0$ out of $7$ points for it, however I fail to see why its wrong: Let $a\in S$ be arbitrary. Since $S$ is finite, there has to be some $m ...
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1answer
26 views

Lifting homomorphism when module is direct summand of free module

Let $N,M,M',N'$ be modules such that $F = N \oplus N'$ is a free module. Let further $f: M \rightarrow M''$ be a surjective homomorphism. I would like to show that for any homomorphism $\phi: N ...
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Show that the union of a chain of ideals is an ideal.

Here is my proof: Let $I=I_1\cup\ I_2\cup\ I_3 \cup\ ..... \cup\ I_n$, $a\in I$ and $r\in R$. Then $a\in I_i$ for some $i$ varying from $1$ to $n$. Since $I_i$ is an ideal of $R$, we have $ar\in ...
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Determining the presentation matrix for a module

I am trying to study some module theory using the book "Algebra" by Michael Artin (2nd Edition, to be precise), and I can't really fathom what is written in Section 14.5. Left multiplication by an ...
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1answer
76 views

Is there any efficient algorithm for computing all semigroups of order n?

I found a paper, but here the author solves a bit different problem. My question is: Is there any efficient algorithm for computing all semigroups of order n?
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1answer
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$N(R)$ when $R$ is a P.I. ring

The set of nilpotent elements $N(R)$ of a ring $R$ with identity is not necessarily a right ideal (or even a subgroup) as it is seen in the ring of $2×2$ matrices over $\mathbb Z$. But, my question ...
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1answer
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Show that $U_{14}\cong U_{18}$?

Because $|U_{14}|=|U_{18}|$ and both of them is cyclic and commutative, so i just need define a function $f : U_{14}\to U_{14}$ that f is homomorphism and bijective. $f(1)=1, f(3)=5, f(5)=7, f(9)=11, ...