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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Field of fractions of a polynomial ring over a ufd [duplicate]

I've a very simple question: if $R$ is a unique factorization domain then what is the field of fractions of the polynomal ring $R[X]$? Any answers or references will be much appreciated!
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Projective covers of graded modules.

I want to prove that there exist projective covers in the category of graded modules over an algebra. i am fairly new to "this" kind of mathematic and don't really know where to start: I found the ...
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1answer
46 views

Prove that $f(x)=x^4+8x^3+x^2+2x+5$ is irreducible in $Q[x]$

Prove that $f(x)=x^4+8x^3+x^2+2x+5$ is irreducible in $Q[x]$ (Q=rationals). I've tried many methods: 1) Eiseinstein's criterion doesn't apply here. I've tried to project the polynom over ...
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1answer
14 views

Centralizer of unique cyclic subgroup of order equal to exponent of group 2

I rewrite my question in a better way. Let $G$ be a finite group and $K\leq G$ the unique cyclic subgroup of $G$ with $|K|=exp(G)$, where $exp(G)=lcm\{|g|\big|g\in G\}$. Is $K=C_{G}(K)$?
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24 views

Centralizer of unique cyclic subgroup of order equal to exponent of group [on hold]

Let $G$ a finite group and $K\leq G$ the unique cyclic subgroup of $G$ with $|N|=\exp(G)$. Is $C_{G}(N)=N$?
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41 views

If a generated subgroup is cyclic

I would like to make a similar question to question "Exercise on generated subgroup": Let $G$ be a finite group and $H\leq G$, $H$ cyclic with $|H|=exp(G)$. If $x\in C_{G}(H)\smallsetminus H$, then ...
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3answers
29 views

How to find the order of a permutation?

Given that $x= (1 2 3)(4 5 6 7 8)$, what is the order of $x$? (i.e the smallest integer $k$ such that
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Semisimple algebras

Let TLn(a) be the Temperley Lieb algebra, How can we know if this algebra is simple or semi simple? I know that if the Jacobson radical J(A)=0 then A is semi simple. But how I find the maximal ideal ...
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Ideal generated by a subset of ring.

The definition of Ideals generated by a subset : Let $S$ be any subset of ring $R$ then an ideal $I$ of $R$ is said to be generated by $S$ if : (1) $S \subseteq I$. (2) for any ideal $J$ of $R$ ...
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2answers
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arranging the variables around when using inverse

If I want to show that $a$*$x$*$a^{-1}$ = $y$, is it acceptable to show that $x$*$a$*$a^{-1}$ = $y$ which then simplifies to $x$*$1$ so $x$=$y$? If not, how could I reorder them using what property?
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Intersection of any family of subfields is itself a subfield

Prove that the intersection of any family of subfields is itself a subfield. In the countable case: Suppose that $\mathscr K$ is a field and consider $(\mathcal K_n)_{n\in\mathbb N}\subset \mathscr ...
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24 views

Semi-Direct Product Proof

I would like to prove that if $G \cong N \rtimes H $ then $N \cap H = \{e\}$. Is it enough to say that if $a \in N$ and $a \in H$, then in $G$ we have $[a,e_h]$ and $[e_n,a]$ - which contradicts ...
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0answers
21 views

Module over an associative ring with unity and axioms of projective geometry

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
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Identifying Symbols [on hold]

When you see $x$ written on a piece of paper you automatically identify it. When you yourself write $x^2 + 2x = 0$ The $x$ you write in $x^2$ differs from the $x$ you write in $2x$ just by a ...
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1answer
19 views

Group of automorphism [on hold]

Let $G$ be group of $(2p)th$ root of unity where p is prime.The group of automorphism of G is 1.cyclic group of order p-1 2.cyclic group of order 2p 3.cyclic group whose order is neither 2p nor p-1 ...
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If $ x^2=e$ for all $x\in G$ then $G$ must be [on hold]

If $ x^2=e$ for all $x\in G$ then $G$ must be cyclic non-abelian abelian finite group.
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In a finite group, the equation $x^m=e$ has $m$ solutions for each positive $m$ that divides the order of the group

Show that in a finite group $G$ or order $n$ written multiplicatively the equation $x^m=e$ has $m$ solutions $x\in G$ for each positive $m$ that divides $n$ I am having trouble understanding how ...
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19 views

Neutral element of an Algebraic structure

Consider $(\epsilon,*)$ an algebraic structure. If the neutral element of $(\epsilon,*)$ is $e$ then it is unique.
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1answer
16 views

Closed Binary Operation over the set G

If * is a binary closed operation, associative and commutative over the set $G$ then * is also a closed operation over the set $H = \{g \in G: g*g=g\}$ My try: $\forall a,b,c \in G $ by hypothesis ...
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Consequence of Lemma: If G is abelian with exponent n, then $|G|\big\vert n^m$ for some $m\in N$

Lemma: If G is abelian with exponent n, then $|G|\big\vert n^m$ for some $m\in N$. Theorem to be proved: Suppose G is finite abelian and group of order m, let p be a prime number dividing m. Then G ...
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1answer
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Why $D_8$ is not primitive as a permutation group on the four vertices of a square?

Why $D_8$ is not primitive as a permutation group on the four vertices of a square? By the way, here is the definition of primitive.
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1answer
45 views

1) Show that $(\Bbb Z[\sqrt2]^*, .)$ is infinite.

1) Show that $(\Bbb Z[\sqrt2]^*, .)$ is infinite. 2) Classify $(\Bbb Z[\sqrt2]^*, .)$, where $\Bbb Z[\sqrt2]^*$ is the group of units of $\Bbb Z[\sqrt2]$ What I have done so far that for $a+b\sqrt2$ ...
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1answer
26 views

f is surjective ⇐⇒ $∀V ⊂ Y$, $f(f^−1 (V ))$ = V prove?

$f$ is surjective ⇐⇒ $∀V ⊂ Y$, $f(f^{−1} (V ))$ = $V$ This is an assertion and i said it was true. But i am confused as to what is referred to as the domain and range in this question. I would say ...
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1answer
23 views

The only subgroup $S$ of $\mathbb{Q}\times \mathbb{Q}$ with $|\operatorname{Aut}(S)|=2$ is $S\simeq\mathbb{Z}\times \{a\}$

An exercise from a lecture: find example of subgroup $S$ of $\mathbb{Q}\times \mathbb{Q}$ with only two automorphisms, i.e., $|\operatorname{Aut}(S)|=2$. Any mistakes? The claim is that the only ...
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1answer
17 views

Show that if $V$ is isomorphic to $A/I$ for some left ideal $I$, then $V$ is a cyclic representation of $A$ over $k$

Suppose we have a representation $V$ of an algebra $A$ over a field $k$. Now assume that there exists a left ideal $I$ in $A$ such that $V$ is isomorphic to $A/I$. Now I have to show that $V$ is a ...
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1answer
18 views

How to find orbits and isoropy group?

About this problem ${a}$, I am wondering if there are 5 orbits in $A$? The 5 orbits separately contain elements which 3 are all the same, 2 of 3 are the same and all 3 are different? I am confused ...
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51 views

Group $G$ such that there is a proper subgroup containing every other proper subgroup of $G$

Characterize all the groups $G$with the following property: There is a proper subgroup $H$ of $G$ such that $\forall S$ proper subgroup of $G$, $S \subset H$. I am pretty lost with this exercise. If ...
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How do I show (G,*) is a group?

Let G be a nonempty set and let * be an associative binary operation on G. Assume that for any elements a,b in G, we can find x,y in G such that a*x=b and y*a=b. (I need to use this assumption. I've ...
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$R$ is a commutative integral ring, $R[X]$ is a principal ideal domain imply $R$ is a field

I've just read a proof of the statement: Let $R$ be a commutative integral ring. If $R[x]$ is a principal ideal domain, then $R$ is a field. In one part of the proof there is a step which I don't ...
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1answer
52 views

Diagonalizing $xyz$

The quadratic form $g(x,y) = xy$ can be diagonalized by the change of variables $x = (u + v)$ and $y = (u - v)$ . However, it seems unlikely that the cubic form $f(x,y,z) = xyz$, can be ...
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Does the binary operation $m ⋆ n = m^n$ on N have a neutral element?

Does the binary operation $m ⋆ n = m^n$ on N have a neutral element? I said yes and it is 1 such that $m ⋆ e = m^e = m$ but apparently that is wrong
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1answer
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How to find the kernel of a transitive action?

I am not sure about this problem. I know $gG_ag^{-1}$ belongs to $G_{ga}$ because $ gG_ag^{-1}(ga)=gG_aa=ga$, but how to prove $G_{ga}$ belong to $gG_ag^{-1}$? What is more, I have no idea about ...
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1answer
22 views

Group of order $|G|=pqr$, $p,q,r$ primes has a normal subgroup of order

Let $p,q,r$ be positive primes, $p<q<r$, and let $G$ be a group with $|G|=pqr$. Show that there exists a normal subgroup $H$ of $G$ of order $qr$. I've seen this post Groups of order $pqr$ and ...
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Give a $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$

Consider $SL_{2}(\Bbb Z_p)$ if q & p be two primes, $p>q$. Give an example of a subgroup $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$ when i) $q|(p-1)$ ii) $q|(p+1)$
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Conjugacy classes in non-solvable group

Suppose $A$ is an arbitrary subset of group $G$ and $K_G(A)$ be the number of $G$-conjugacy classes contained in $A$. Also suppose $G$ non-solvable group, $N\unlhd G$, $G/N$ is abelian, $|G/N|=6$ and ...
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1answer
42 views

First Isomorphism Theorem to identify a quotient

I understand the notion of quotient groups quite well (I think), but I'm struggling a little bit with the following problem: Let $G$ denote the group of 2x2 invertible real upper triangular ...
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Conjugacy classes in non-abelian simple group

Can we say that every non-abelian simple group has at least 4 non- identity classes?
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1answer
18 views

For a linear function, the fiber of the output is the translate of the kernel by the input. (Trivial observation, proof needed.)

As you may already know, I am a newbie to linear algebra. I am supposed to prove that for every linear function between vector spaces, for every input, the fiber of the corresponding output equals the ...
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24 views

How can we prove that every maximal ideal is a prime ideal? [on hold]

In abstract algebra,how can we prove that every maximal ideal is a prime ideal? Give full logical proof.
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Endomorphism rings of a $k$-algebra

Let $k$ be a field and $R$ be the $3$-dimensional $k$-algebra $\left [\begin{array}\ k & k \\ 0 & k \end{array} \right ]$. I have two conjectures: (1) Is any simple right $R$-module has ...
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Simplest way to show that $A_n$ is simple for all $n \geq 5$?

What's the simplest and shortest proof to show that $A_n$, the alternating group of $S_n$, is simple for all $n \in \mathbb{Z}$, such that $n \geq 5$?
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Can we say “commutative ring = field”?

We know the difference between ring (R) and field (F) is that R does not guarantee multiplication is commutative. Now, if considering commutative R, which means (R,.) is a group, can we say: ...
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1answer
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Isomorphism classes of abelian groups of certain order

working on a practice question about finite abelian groups and just want to see if I am on the right track: Let $H = <(123)(4567),(8\space 9)(10\space 11),(8\space 11)(9\space 10) > \space ...
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1answer
34 views

Colon ideal of fractional ideals is itself a fractional ideal

I received this question on homework in my homological algebra class and I need some guidance. Let $R$ be a commutative integral domain and $K$ be its field of fractions. A fractional ideal $I$ of ...
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The number of elements in $\mathbb{Z}_{11}$ satisfies $x^{12}-x^{10}=2$. [duplicate]

The number of elements in $\mathbb{Z}_{11}$ satisfies $x^{12}-x^{10}=2$. I don't know how to start it.
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The number of subgroups of $\mathbb{Z}_3\times \mathbb{Z}_{16}$

I want to calculate the number of subgroups of $\mathbb{Z}_3\times \mathbb{Z}_{16}$. But it is just that to calculate the number of subgroups of $\mathbb{Z}_3$ and $\mathbb{Z}_{16}$. It is easy to ...
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1answer
43 views

Algebra + Real Analysis video lectures

I'm an undergraduate taking graduate courses beginning a research project. I don't have much time but want to brush up on my Algebra and real analysis at a graduate level. Does anybody know any good ...
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1answer
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Hilberts Theorem (norm group)

The theorem says the following: The map $N$ is a group homomorphisim from the multiplicative group of $\mathbb{Q}^{x}[i]$ to the multiplicative group of $\mathbb{Q}^{x}$ and has kernel $\lbrace ...
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Does there exist a group in which this property does not hold? [on hold]

Let $g, h \in G$ is there a group where $(gh)^n \neq g^nh^n$?