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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How to show that $3\mathbb{Z}/15\mathbb{Z} \cong \mathbb{Z}/5\mathbb{Z}$?

How to show that $3\mathbb{Z}/15\mathbb{Z} \cong \mathbb{Z}/5\mathbb{Z}$ as $\mathbb{Z}$-module over $\mathbb{Z}$? My proof: Define surjective function $f:3\mathbb{Z}/15\mathbb{Z} \rightarrow ...
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27 views

Inverse property for groups Proof

I was wondering if (1) this proof is correct, and (2) if other proofs exist for the following: Prove that $(a_1a_2...a_n)^{-1}=a_n^{-1}a_{n-1}^{-1}...a_1^{-1}$ where $a_i \in $ a Group $G$ Proof by ...
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Image of homomorphism with kernel $N$ isomophic to $M/N$

Let $M$ be an $R$-module and $N$ be a submodule. Let $f:M\rightarrow M'$ be a surjective homomorphism with kernel $N$. How to show that $M'$ is isomorphic to $M/N$? My thought: Define ...
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38 views

Prove that $a+ib$ is prime in $\Bbb Z[i]$, of $a^2+b^2$ is prime in $\Bbb Z$.

Prove that $a+ib$ is prime in $\Bbb Z[i]$, of $a^2+b^2$ is prime in $\Bbb Z$. My Try: We can easily show that $\Bbb Z[i]$ is a FD but how can we show that $\Bbb Z[i]$ is a UFD. Because if we can show ...
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12 views

Simple composition factor

We know that if $e$ is a primitive idempotent in a semiperfect ring then $Re/Je$ is a simple left $R$-module, where $J$ is the Jacobson radical of $R$. Now, suppose that $e$ is an idempotent in an ...
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1answer
19 views

To show module homomorphism being injective

Suppose $R$ is a field, $M$ is a right $R$ module, and $f : R_R \rightarrow M$ is a non-zero homomorphism. Show that $f$ is injective. My work: The homomorphism is uniquely determined by $f(1)$. If ...
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2answers
36 views

Show that $2$ is not prime in $\mathbb Z[\sqrt{-d}]$ for odd prime $d$

I have to show that if $d > 2$ is a prime number, then $2$ is not prime in $\mathbb Z[\sqrt{-d}]$. The case when $d$ is of the form $4k+1$ for integer $k$ is quite easy: $2\mid ...
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27 views

given polynomial has a root in $Z_p$…

To check $f_x$ is irrudicble or not in $f_p$ check wheather 0,1,2 , p-1 is a root of $f_x$...if any of this is a root then it is not irrudicible.. is this method applicable here?
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1answer
28 views

How many possible isomorphisms do we have between G and H? [duplicate]

Let $G=(Z_4,+)$ and let $H=(U_5,*)$ where $U_5 = \{[1],[2],[3],[4] \}$ . I know that $[1]$ and $[3]$ are both generators for $G$. I also know that $[2]$ and $[3]$ are both generators for $H$. In order ...
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23 views

Should a reflection matrix of a vector have the same form as a rotation matrix?

According to the book: I know that it is not possible to write a reflection as a rotation, but from the text it seems that the matrix of the form $$ A=\left[ \begin{array}{ c c } a & ...
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2answers
20 views

The abelianness of the quotient group of an abelian group.

I am working on an assignment for my abstract algebra class. The question states: Let $A$ be an abelian group and let $B$ be a subgroup of $A$. Prove that $A/B$ is abelian. I was under the ...
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38 views

Isomorphisms between $(\mathbb Z_4,+)$ and $(U_5,*)$

So I am asked to find all the isomorphisms between $G = (\mathbb Z_4,+)$ and $H = (U_5,*)$. I solved it as follows: we will have two isomorphism corresponding to the two generators of $U_5$. The ...
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13 views

Find up to isomorphism all the quotient groups of composition series of a group of order $30$.

I can't seem to understand what I should do here... All I did so far is proving that $G$, (such a group), is not simple. But there are many cases, I can't really tell what $n_2,n_3$ and $n_5$ are, ...
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27 views

Is my observation correct regarding restriction of scalars?

Let $\alpha: \Lambda\to \Gamma$ be a ring homomorphism, then $ _\Lambda\Gamma_\Gamma$ is a bimodule. We have the following pairs of adjoint functors $$ \mathbf{Mod_\Lambda} \xrightarrow{\cdot\; ...
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1answer
37 views

Is there a formalism for a universal mathematical representation of algorithms?

I don't know if my question is correct so excuse me if I'm not 100% clear about what I would want to know. Is there a formalism which can capture all possible algorithms (mathematically speaking) ? ...
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2answers
23 views

Finding an order of a coset in $A/B$ where $A$ is a free abelian group and $B$ is a subgroup.

Let $A$ be a free abelian group with basis $x_1,x_2,x_3$ and let $B$ be a subgroup of A generated by $x_1+x_2+4x_3, 2x_1-x_1+2x_3$. In the group $A/B$ find the order of the coset $(x_1+2x_3)+B$. How ...
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1answer
14 views

Group Z, find<m,n>

How to find <8,14>, in group Z under addition, Any integer k such that the subgroup is . For 8, <8>={8n : n$\in$Z}, <14>={14n:n$\in$Z}, so <8,14>={22n and 6n:n$\in$Z}, is it right? how to ...
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42 views

Commutative generators of a group

If a group has commutative generators is the group always abelian? I have a question dealing with how to determine if a Cayley graph of a group is an abelian group. It seems that if the generators ...
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6answers
135 views

Prove that $x-1$ is a factor of $x^n-1$

Prove that $x-1$ is a factor of $x^n-1$. My problem: I already proved it by factor theorem† and by simply dividing them. I need another approach to prove it. Is there any other third ...
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2answers
49 views

How many homomorphisms are there from $\Bbb{Z}_6 \to \Bbb{Z}_{18}$?

I need to determine how many homomorphisms there are from $\Bbb{Z}_6 \to \Bbb{Z}_{18}$. I have never solved that kind of question. I do know that orders are preserved and that some elements can be ...
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3answers
55 views

Are the $2\times 2$ symmetric matrices a ring?

Ok so I am looking at Rings. I saw somewhere that the $2 \times 2$ symmetric matrices with entries in $\mathbb{R}$ is a ring. But if we look at matrix multiplication I am not convinced: If $ A = ...
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2answers
228 views

Example of non-Abelian group with 4, 5, or 6 elements of order 2

Is there any example for non-Abelian group in which has more than $3$ elements of order $ 2$, but has less than 7 elements of order $2$?
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Multiple products of submodules

NOTE: This is part of a homework, so only worry about the question regarding notation. We have the following conditions: $R=\mathbb{Z}$, $I = \mathbb{Z}_{>0}$, and $M_i = \mathbb{Z} / i ...
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0answers
25 views

Algebra A over a field F contains no non-trivial left F-ideals if and only if A contains no non-trivial right F-ideals [on hold]

Algebra $A$ over a field $F$ contains no non-trivial left $F$-ideals if and only if $A$ contains no non-trivial right $F$-ideals. Why this fact is true? Or is it true? I think it's easy thing, ...
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1answer
31 views

Galois group of $X^5-1\in\mathbb F_7$

I want to find the Galois group of $X^5-1$ over the finite field $\mathbb F_7$ but I don't know how to find Galois groups over finite fields. Over $\mathbb Q$ the Galois group $\text{Gal}(\mathbb ...
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1answer
21 views

All automorphisms of splitting fields

Let $M \le \mathbb{C} $ be the splitting field of polynomial $ f(x) \in \mathbb{Q}[x] $. Find all automorphisms of field $ M $ in cases: 1) $ f(x) = x^6 - 1 $ 2) $ f(x) = x^{2011} - 1 $ 1) In ...
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2answers
30 views

Relationship between submonoids and subgroups

I'm taking my first abstract Algebra course. During one of the last lessons, my teacher told us that If $G$ is a group and $M$ is a finite submonoid of $G$, then $M$ is a subgroup of $G$. For the ...
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45 views

question regarding group theory proof

Can someone please explain the sentence in red?, how does it follow?
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2answers
34 views

Criteria for the existence of zero-divisors and idempotent elements in the integers modulo $n$

I need help in establishing or at least deciding the validity of the following two criteria: There are in the ring $Z_n$ non-trivial zero divisors if only if $n$ is divisible with some square. ...
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1answer
37 views

Elliptic curve-point at infinity

In my lecture notes we have the following: $$P \oplus Q \oplus R =O \Leftrightarrow P, Q, R \text{ are collinear }$$ So $$P \oplus Q \oplus O =O \Leftrightarrow Q=-P$$ that means that $Q=-P$ ...
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Bigenetic properties of finite group [on hold]

Nilpotency, supersolubility and polycyclicity are bigenetic properties of the class of all finite group. Let be : P is property, X be a class o group. We say that P is a bigenetic property of ...
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1answer
57 views

Irreducible Polynomial in $\mathbb{Q}[x]$ [on hold]

Show that $x^4+2x^2+4$ is irreducible in $\mathbb{Q}[x]$.
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1answer
51 views

Automorphism group of the ring $\mathbb{F}_3\left[t,\frac{1}{t}\right]$

Let $R=\mathbb{F}_3\left[t,\frac{1}{t}\right]$ be a ring. What is the simplest form of $\mathrm{Aut}(R)$ ? Here $t$ is a variable and $R$ is the smallest ring contained in field ...
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21 views

Can a strict inequality be derived from a weaker one?

Suppose P and Q are two statements, with P being the stronger one. Let us denote the set of statements derived from P and Q be A and B respectively. Then can the strongest statement belonging to A be ...
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1answer
22 views

Properties of Jacobson radical

I'm looking for an example of ring epimorphism $\varphi:R\rightarrow S$ such that the natural homomorphism $\tilde\varphi:J(R)\rightarrow J(S)$ is not surjective, where $J(R)$ is a Jacobson radical.
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2answers
34 views

Computing a Galois group

Let $K=\mathbb Q$ and $L$ be the splitting field of the polynomial $f(x)=x^4-6x^2+4$. I want to calculate the Galois group $\text{Gal}(L/K)$. The zeros of $f$ are obviously ...
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1answer
25 views

Generators over semiperfect rings

It is clear that if $R$ is a ring with identity and $e\in R$ is an idempotent then $Re$ is a direct summand of $R$ while $R$ is a generator in the category of left $R$-modules. I have my question when ...
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2answers
82 views

prove or disprove $H$ is a subgroup

If $H$ is a nonempty subset of a group $G,$ and if $a,b\in H,$ then $a^{-1}b^{-1} \in H,$ can we prove that $H$ is a subgroup of $G$? if not, how to disprove it?
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1answer
30 views

What are the irreducible representations $V$ for $S_n$ over ${\bf C}$ that admit a nonzero vector fixed by $S_{n-1}$? [on hold]

Find with proof all irreducible representations $V$ for $S_n$ over ${\bf C}$ that admit a nonzero vector fixed by $S_{n-1}$.
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2answers
33 views

Find order of polynomials in a group [on hold]

$G$ is the group of polynomials under addition with coefficient from $\mathbb Z_{10}$. How to find the order of $f(x)=7x^2 + 5x+4$? If $h(x)=a_nx^n+...+a_0 \in G$, how to get the order of $h(x)$ if ...
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1answer
28 views

The dual of a tensor algebra as a right module

Let $V$ be a finite-dimensional vectorspace over a field $k$, and let $T$ denote the tensor algebra of $V$ (thought as a graded $k$-algebra). Denote by $T^\vee$ the dual of $T$, i.e. ...
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2answers
72 views

Order of any element divides the largest order.

Let $A$ be a finite Abelian group and let $k$ be the largest order of elements in A. Prove that the order of every element divides $k$. This is my attempt, I sense there is something wrong\incorrect ...
2
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1answer
25 views

$V^{\oplus3}$, linear constraints. [on hold]

Let $V$ be an irreducible $G$-representation over $\mathbb{C}$, and let $W = V \oplus V \oplus V$. Prove that all submodules of $W$ are given by "imposing linear constraints," e.g.$$\{(x, y, z) \in V ...
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2answers
53 views

Verify that R is a ring

Let $\alpha = \frac{1}{2}(1+\sqrt{-19}) \in \mathbb{C}$ and $R = \{a+b\alpha\mid a,b \in \mathbb{Z}\} \subseteq \mathbb{C}$. Verify that $R$ is a ring under usual addition and multiplication. My ...
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3answers
59 views

Simple form of a ring

What is a simple form of this ring: $$\mathbb{Z}[\sqrt{2}][x]/(5,x^2+1),$$ I know that $\mathbb{Z}[\sqrt{2}][x]=\mathbb{Z}[x,y]/(y^2-2)$. Probably, I should use second theorem of isomorphism, but I ...
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28 views

When is the centralizer and the center are equal

Hey guys so we always know the centralizer is always part of the center but a centralizer isn't necessarily equal to the centralizer what restrictions can we impose on the group in order for them to ...
3
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0answers
46 views

Are the following maps linear?

A linear map $T:V\rightarrow W$ is a function satisfying: $T(v_1+v_2)=T(v_1)+T(v_2), \forall v_1,v_2\in V$ $T(\alpha\cdot v_1)=\alpha\cdot T(v_1), \forall \alpha \in \mathbb F$ I am unsure if I ...
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32 views

$V$ is $G$-irrep. over $\mathbb{C}$, submodules of $V \oplus V \oplus V$ given by imposing linear constraints. [on hold]

Let $V$ be an irreducible $G$-representation over $\mathbb{C}$. Let $W = V \oplus V \oplus V$. Show that all submodules of $W$ are given by "imposing linear constraints," e.g.$$\{(x, y, z) \in V ...
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1answer
30 views

Modified first isomorphism theorem

This question is vague on purpose but I hope it should make enough sense. I don't think it matters which type of objects I'm working with, but just in case it does, I'll point out that I'm working ...
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0answers
37 views

Is this the correct way to calculate the matrix of change of basis?

I've attempted to solve the following problem: Let V be the subspace of $C[0,1]$ spanned by the vectors of the linearly independent sequence $B=(e^x,xe^x,x^2e^x)$. Let D be the differentiation ...