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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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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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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1answer
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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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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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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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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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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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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
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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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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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1answer
30 views

$R$ 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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46 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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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
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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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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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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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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
29 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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2answers
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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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1answer
28 views

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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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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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 $(gh)^n \neq g^nh^n$?
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Find the sum of the $9$th powers of the roots of $x^3+3x+9=0$

Let $$x^3+3x+9=0$$ and $x_1,x_2,x_3$ be the roots of this equation. Given that $\displaystyle S=x_1^9+x_2^9+x_3^9$. What is the exact value of $S$? I think that $S=0$, but I am not sure. I tried to ...
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4answers
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A stimulating book about algebra

I need to fill some gaps in my algebra knowledge. The problem is: While I do realise the importance and utility of the subject, I do not find it appealing. Is there any book around which shows ...
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1answer
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Module over a ring which satisfies Whitehead's axioms of projective geometry

I asked this question(Characterization of a vector space over an associative division ring). It was pointed out that the answer was negative. So I reconsidered the problem and have come up with the ...
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1answer
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Map from $\mathbb{R}\mathbb{P}^{2}$ to Klein bottle homotopic to constant map

I'm currently struggling with the following exercise: Show that any continuous map $$f: \mathbb{R}\mathbb{P}^{2} \to K$$ where $K$ is the Klein bottle, is homotopic to a constant map. I know that ...
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Semi direct product

Prove that (i) $GL_n(R)= \coprod_{w\in S_n} UwB$ where $w \in S_n$ is a permutation matrix. and $U$ is a subgroup of $GL_n(R)$ consisting of upper triangular matrices with diagonal entries $1$ and ...
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1answer
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Algebra - proof verification involving permutation matrices

Theorem. Let $\textbf{P}$ be a permutation matrix corresponding to the permutation $\rho:\{1,2,\dots,n\}\to\{1,2,\dots,n\}$. Then $\textbf{P}^t=\textbf{P}^{-1}.$ Proof. First note the following ...
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1answer
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How to show that $\mathbb{C}[x_1,x_2,x_3, x_4]/(x_1x_2 - x_4x_3, x_1x_3 - x_2x_4, x_4x_1 - x_3x_2)$ is integral domain

I am looking for a way to show that the ring $\mathbb{C}[x_1,x_2,x_3, x_4]/I$ where $I = (x_1x_2 - x_4x_3, x_1x_3 - x_2x_4, x_4x_1 - x_3x_2)$ is an integral domain. In other words I want to show $I$ ...
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2answers
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Subgroups and justification

Which of these subsets are subgroups of the given group and justify your answer. The group $R^+$ of postive reals under multiplication. The subset $H=(3n|$ $n\in Z^+)$. The group of nonzero ...
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Groups of Order 2 with subgroups

Let G be an abelian group and $a,b\in G$ be two distinct elements with a and b or order $2$. Show that $H=\{e,a,b,ab\}$ forms a subgroup and write out its multiplication table. Justify why all the ...
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stalk of projective variety in terms of the coordinate ring

Let $X \subseteq \mathbb{P}^n$ be an embedded projective variety over some field $k$ with its corresponding homogeneous coordinate ring $R = k[X_0,\dots,X_n]/I(X)$. Let further $X = \bigcup_{i=0}^n ...
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How can I show that the tensor product of $\mathbb Z$ and $\mathbb R$ as $\mathbb Z$-modules is isomorphic to $\mathbb C$? [on hold]

How can I show that $\mathbb Z\otimes_{\mathbb Z}\mathbb R\simeq\mathbb C$ as $\mathbb Z$-modules? I'm unable to come up with a solution as I'm quite new to Commutative Algebra. Can someone ...
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2answers
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Prove that, if $(a)=(a')$, then $a'=ua$

Let $R$ be integral domain. Show that if $2$ principal ideals $(a)$ and $(a')$ are equal (where $a,a'\in R$) then there exists $u\in R^{\times}$ such that, $a'=ua$ Now if $(a)=(a')$ then ...
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1answer
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When can a group be decomposed into a direct product of smaller groups?

Is there any general condition that a group must satisfy in order to be decomposable into a direct sum or product of smaller groups? And what happens if one replaces 'direct product' with semi-direct ...
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3answers
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Every non-unit is in some maximal ideal

I am trying to prove that every non-unit of a ring is contained in some maximal ideal. I have reasoned as follows: let $a$ be a non-unit and $M$ a maximal ideal. If $a$ is not contained in any maximal ...
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1answer
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Whitehead's axioms of projective geometry and a vector space over a field

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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1answer
55 views

Proof about isomorphism?

Who has clues about this problem? It is so difficult for me. THANKS A LOT!
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1answer
31 views

Characterization of a vector space over an associative division ring

Let $M$ be a (left) module over an associative division ring $R$. Then it has the following properties. 1) For every submodule $N$ of $M$, there exists a submodule $L$ such that $M = N + L$ and $M ...
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$H^1(G,A)\rightarrow H^1(H,A)$ is onto

Note that $H^1(G,A)\rightarrow H^1(H,A)^{G/H}$ where $A$ is $G$-module and $H$ is a subgroup in $G$ But I suspect that ${\rm Res}\ : \ H^1(G,A)\rightarrow H^1(H,A)$ may be onto since $ f\in C^n ...