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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A question about commutator subgroup

let $G=H×K$ be a nonabelian group that $o(H) = p^2$ , $o(K) = p^3$ (p is prime number), why is $o(G') = p$?
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1answer
108 views

What is the automorphism group of $\mathbb P^1 \times \mathbb P^1$?

I know that the automorphism group of the projective space $\mathbb P^n$ is $PGL(n+1)$. What is it for $\mathbb P^1 \times \mathbb P^1$? Apart from individual automorphisms of the factors $\mathbb ...
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1answer
150 views

'ED implies UFD' proof without PID?

Is there any way to prove 'ED implies UFD' without using the idea of PID? The proof I know is the one in basic algebra books, the one that uses PID. I admit that the introduction of PID makes ...
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1answer
129 views

Severe problems understanding abstract algebra [closed]

I just came out of high school and one of my courses during my first semester of a maths B.sc. is Abstract Algebra. Now all my other courses seem logical and even easy, but this course...I just don't ...
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2answers
48 views

Showing that an epimorphism to a free module of finite rank splits

Let $M$ be an $R$-module and let $F$ be a free $R$-module of finite rank. Let $\phi : M \to F$ be an epimorphism. Then show that $M$ has a submodule $F' \cong F $ such that $M=F' \oplus \ker\phi$. ...
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44 views

If there is an integer $n$ such that $n^2\equiv3\pmod p$, where $p$ is prime, prove there are integers $a$ and $b$ such that $|a^2-3b^2|=p$

If there is an integer $n$ such that $n^2\equiv3\pmod p$, where $p$ is prime, prove there are integers $a$ and $b$ such that $|a^2-3b^2|=p$. So $n^2-3 = pm$ for some integer $m$, and I know $|a^2 ...
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1answer
476 views

Group of order $63$ has an element of order $3$, without using Cauchy's or Sylow's theorems

I'm showing that any group of order $63$ has an element of order $3$, and can only use Lagrange's theorem not Cauchy's or Sylow's. I got it reduced to a case of having $62$ elements of order $7$ but ...
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1answer
80 views

Find a maximal ideal $I$ in the ring $\mathbb{Z}[i]$ such that $\mathbb{Z}[i]/I$ is isomorphic to $\mathbb{Z}/521\mathbb{Z}$.

I know $\mathbb{Z}[i]$, the Gaussian integers, is a PID. So $I$ is generated by a single element. At first I thought $I=(521)$, but $521$ can be reduced to $11^2 + 20^2$. Would $I=(11 + 20i)$ or ...
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0answers
66 views

Group of invertible elements, isomorphic to $Z_4$?

The group $f(8)$ of invertible elements in the ring $Z_{10}$ has four elements, $f(10) = \{ [1,], [3], [7], [9]\}$. Is this group isomorphic to $Z_4$ or to the symmetry group of the rectangle? ...
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56 views

modular arithmetic proof

Suppose $x$, $y$, and $z$ are integers and $x= 3y^2 -z^2$. Prove that $x\not\equiv1\mod4$. My thoughts: So I am not sure the route that can prove this. I am trying to just use the simple stuff to ...
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2answers
52 views

Simple Lie algebra is a matrix algebra?

Wedderburn's Theorem. Let $A$ be a simple finite $k$-algebra. Then $A$ is a matrix algebra over a finite $k$-algebra $K$ which is a skew field. (Here matrix algebra means $A=M_n(K)$ for some $n$.) ...
2
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1answer
84 views

Are these two theorems about algebraic varieties the same?

In Artin's Algebra, there is a theorem (1) stated as the following: Let $J\subset\Bbb{C}[x]$ be an ideal such that $J=(f_1,\cdots,f_r)$ where $f_1,\cdots,f_r\in\Bbb{C}[x_1,\cdots,x_n]$. Let ...
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0answers
74 views

simple question regarding isomorphisms relating to ring of integers

I have a simple question about isomorphisms and ideals. Let $\mathcal O_F$ be the ring of integers in some quadratic number field $F=\mathbb{Q}(\sqrt d)$ and let $f(x)$ be the minimal polynomial of ...
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1answer
127 views

How to compute Ext over an exterior algebra

I found this question in several places (even on mathoverflow and mathstackexchange), but I never found a satisfying answer. Let $k$ be a field and $V$ a finite dimensional $k$-vectorspace. I ...
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2answers
68 views

Subring of $\Bbb Z_{18}$ with unity

Need help finding subrings $A$ and $B$ of $\Bbb Z_{18}$ in which $A$ and $B$ are rings with unity, $B$ is a subring of $A$, but the unity of $B$ is not the same as the unity of $A$.
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635 views

What is the intersection of all Sylow $p$-subgroup's normalizer?

Intersection of all Sylow $p$-subgroups is generally denoted by $O_p(G)$ and it is one of the well studied topics in group theory as there are many theorems related to this. Let $R$ be ...
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1answer
67 views

Why is this is an equivalence relation?

Fulton makes the following definitions: After he defines an equivalence relation: The definitions he made seems very obscure to me and if anyone could show why this relation is an equivalence ...
2
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1answer
77 views

Existence of Homomorphism

Given two homomorphisms $\alpha$ : A $\rightarrow$ B and $\beta$ : A $\rightarrow$ C, if $\ker( \beta ) \subseteq \ker( \alpha )$ and $\beta$ is onto (surjective), show that there is a homomorphism ...
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49 views

induction on power elements in a group

Exercise: Prove by induction the following facts about power elements in a group. For all integers $k$ and $l$, $(a^k)^{l} = a^{kl}$ I am having trouble with induction on two integer variables. ...
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3answers
107 views

solve $x^{15} \equiv 2\pmod {47}$

solve $x^{15} \equiv 2\pmod {47}$ on the solution set: suppose that $\overline{x^{15}} = \overline{2}$ then $\overline{x^{15m}} = \overline{2^m}$ solve $15m \equiv 1 \pmod {46}\;$, gaining a ...
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1answer
108 views

Subgroup of a cyclic groups and are isomorphic

I don't know whether I am right or wrong. Can anyone help me to clear below problem ? Question 1: Let C2 be the cyclic group of order 2 and C202 be the cyclic group of order 202. Find all subgroups ...
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2answers
59 views

Product of disjoint cycles in Abstract Algebra

I've got some permutations: $\left( \begin{smallmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 3 & 5 & 4 & 1 & 6 & 2 \end{smallmatrix} \right) $ and $\left( ...
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1answer
95 views

Abstract Algebra: Find a number of non-isomorphic subgroup of $S_3$

I'm having difficulty in understanding the method to find the solution for this question. I repeat Question: Find the number of non-isomorphic subgroup of $S_3$ So is this the way to find the ...
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1answer
47 views

Help in this proof on DVRs

I'm trying to understand this proof: Anyone could clarify the converse please? I really need help. Thanks a lot
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3answers
48 views

Easier way to show $(\mathbb{Z}/(n))[x]$ and $\mathbb{Z}[x] / (n)$ are isomorphic

$$(\mathbb{Z}/(n))[x] \simeq \mathbb{Z}[x] / (n)$$ I've shown this by showing that the map that sends $\overline{1} \mapsto [1+(n)]$ (where the bar denotes the congruence class mod $n$) and $x ...
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1answer
60 views

Quotient of $\mathbb{C}[x]$ by $(x-a)^2$ or by $x^2$.

Are $\mathbb{C}[x]/((x-a)^2)$ and $\mathbb{C}[x]/(x^2)$ isomorphic? I see that the elements (cosets) of each rings can be identified with linear polynomials $c_1 x + c_0$, but I am not sure what to ...
5
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1answer
69 views

Why is a normal subgroup of $G_1\times G_2$ with trivial intersections with $G_1$ and $G_2$ is abelian?

Let $G=G_1\times G_2$ be a direct product, and let $H\triangleleft G$ be a normal subgroup such that $H\cap G_1=H\cap G_2=\{1\}.$ Then $H$ is abelian. I considered the commutators of two ...
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1answer
42 views

Two questions about the lattice derived from 0th-order formulas

It's not clear to me if the definitions I've been given are common. Therefore I will give a brief overview of the constructions I'll need to talk about the objects I want to. Prerequisite: Given ...
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353 views

Proof of Divisibility of $n(n^2+20)$ by 48.

This is a question from Bangladesh National Math Olympiad 2013 - Junior Category that still haunts me a lot. I want to find an answer to this question. Please prove this. If $n$ is an even ...
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1answer
43 views

A Problem about Spliting of Exact Sequence

Module Case Let $$0\rightarrow N \stackrel f \rightarrow G\stackrel g \rightarrow Q\rightarrow 0$$ be the exact $R$-module sequence iff (1) there exists $u:G\rightarrow N$ which satisfies $u\circ ...
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0answers
57 views

Algebra trick question: quotient/factor group isomorphic to a subgroup of $S_n$ where $[G:H]=n$ [duplicate]

I think this might be a trick question, even if it is I am still not sure how to write it. Let G be a group Let H be a subgroup of G I wish to show that if $|G:H|=n$ with $n>1$ then G has a ...
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1answer
188 views

Galois group of an irreducible cubic polynomial without using the discriminant

Let $f\in\mathbb Q[x]$ be irreducible of degree $3$. Since the Galois group $G$ of $f$ is a transitive subgroup of $S_3$, it is either $S_3$ or $A_3$. Those two possibilities are easily ...
2
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2answers
54 views

Solvability of an equation

Let $p\left( x\right) =x^{n}+ax+b$ and $a,b>0$. Is the equation $p\left( x\right) =0$ always solvable? Which are the solutions?
2
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1answer
68 views

Central algebra vs simple algebra

Let $k$ be a skew field. Assume that $A$ is finite $k$-algebra, i.e., ${\rm dim}_k A = [A:k] < \infty$. Before asking I will enumerate two definitions : Def : A $k$-algebra $A$ is $central$ ...
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0answers
82 views

'Finitely generated modules' versus 'Finitely generated algebra'

I'm reading Commutative Algebra by Bourbaki, and I'm on chapter III of the book. I really think that the author made a typo there, but I'm not so sure. It's Proposition 1, and its Corollary on page ...
3
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1answer
63 views

Factoring $x^{16}-x$ over $\mathbb{F}_8$

A homework question asks me to factor $x^{16}-x$ over the finite fields $\mathbb{F}_4$ and $\mathbb{F}_8$. I got the result for $\mathbb{F}_4$ using the factoring over $\mathbb{F}_2$ and then a ...
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2answers
101 views

Is the group $G$ cyclic?

Assume that $G$ is a finite group such that for any positive integer $n$ dividing $|G|$, $G$ has one and only one subgroup $H$ with $|H|=n$. Is $G$ cyclic?
2
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1answer
87 views

Commuting matrices still commute if you conjugate one of them?

If two square matrices, $A$ and $B$, of the same size and with complex entries commute, why does $\overline{A}$ (the complex conjugate of $A$) commute with $B$? If this is not true, is it true if one ...
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1answer
27 views

$T_p$ is a group of permutations defined $t(x)=ax+b$ for some $a,b,x \in \mathbb{Z}_p$ where $a \neq 0$

$T_p$ is a group of permutations defined $t(x)=ax+b$ for some $a,b,x \in \mathbb{Z}_p$ where $a \neq 0$ and $p = $ prime. a) Show that $T_p$ is a group with order $p(p-1)$. b) Also prove that ...
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1answer
94 views

Quotient groups and the first isomophism theorem

I've not found a duplicate but there's not much to go on. This is a question I have posed myself after thinking about a question in Sege Lang's "Undergraduate Algebra". The matrices are over the ...
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1answer
110 views

Smallest skew-field containing a non-commutative ring.

Let $R$ be an integral domain and take $D = R - \left\{ 0 \right\}$. The ring $D^{-1}R$ is the smallest field containing $R$ as a subring. Now suppose that I have a non-commutative ring $N$. Suppose ...
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1answer
164 views

A question on valuation overrings of a PID

Let $A$ be a PID and let $K$ be its quotient field. Let $V$ be a valuation ring of $K$ containing $A$ and assume $V\neq K$. Show that $V$ is a local ring $A_{(p)}$ for some prime element $p$. I ...
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1answer
199 views

Number of Non-Isomorphic Subgroups of $S_3$

I am relatively new to abstract algebra, and am having trouble solving this problem. How would I go about finding all these subgroups that are not isomorphic to a permutation group of 3? So far I only ...
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1answer
46 views

What is the order of this group? [duplicate]

Let $H$ be the subgroup of the group $G$ of all $2 \times 2$ non-singular matrices whose entries are integers modulo a given prime $p$ consisting of those and only those matrices in $G$ whose ...
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2answers
102 views

the value of a polynomial of degree 1 over $\mathbb Z[x]$ is divisible by infinitely many primes

Here is a problem from Lang's Algebra,chapter 12 and problem 12, a)Let f(x) be a polynomial of degree 1 in $\mathbb Z[x]$. Show that the values $f(a)$ for $a \in \mathbb Z$ are divisible by ...
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1answer
69 views

Multiplicative identity of a quotient ring

Let $R$ be a commutative ring such that $M$ is a maximal ideal of $R$. Then I know that $R/M$ is a field. But I am unable to understand what is the multiplicative identity of $R/M$? If $R$ has an ...
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1answer
256 views

Normalization of a quotient ring of polynomial rings (Reid, Exercise 4.6)

I solved all parts of Exercise 4.6 of the book Undergraduate Commutative Algebra of Miles Reid except the last one. Let $A=k[X]$ and $f\in A$ has a square factor but it is not a square polynomial ...
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1answer
45 views

Proof that for $x,y\in G$ and $y=xa$ for some $a\in H$ where $H$ is a subgroup of $G$ that $xH=yH$

I want to prove that for $x,y\in G$ and $y=xa$ for some $a\in H$ where $H$ is a subgroup of $G$ that $xH=yH$. I think I am sort of on the right lines but I lack confidence I'd like to have this ...
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1answer
50 views

What's the name of the mathematical structure with is an abstraction of things like linear Independence?

This is a terrible question, I know. I can't remember the details for some reason, but I think (hope?) that anyone who's familiar with this object will immediately know what I'm talking about.
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0answers
84 views

Simplify this expression as much as possible?

I am working on a proof and I am stuck on this problem. I have this expression that I want to simplify this as much as possible but I don't get how. I want to try removing the "$...$" using possibly ...