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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Let $H\subset G, K\subset H$ be normal subgroups such that $K$ is normal in $G$ .Describe $(G/K)/(H/K).$

First of all I would like to say hello to all of you! I am new here! :-) I am very pleased to have found this site. So I can help others if they need me and get help when I need it, really a great ...
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23 views

Definition of direct sum of modules?

I have just started studying modules, and am trying to figure out the definition of the direct sum of modules but I'm having trouble since different sources seem to give different definitions, for ...
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138 views

How to show that there is a bijective correspondence between two sets of prime ideals

I'm trying to solve this Algebra Problem, and I'm not quite sure, if I'm on the right way. Let $R$ be a commutative ring and $S \subset R$ a multiplicative subset. Show that $p \to pS^{-1}R$ ...
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106 views

Is every free subgroup a direct summand?

Let $G$ be an abelian group. Suppose that $F$ is a subgroup of $G$ such that $F$ is free. Does there necessarily exist a subgroup $H\subset G$ such that $G\cong F\oplus H$? Motivation: In Lang's ...
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41 views

Is every symmetric group generated by some cycles?

Actually, I have two questions here: How to show that $D_3$, the dihedral group, is isomorphic to $S_3$ in details? Is every symmetric group generated by some cycles? For question (1), I know ...
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30 views

Group Extension Example

I am trying to find two distinct group extension of $Z_2$ by $Z_3$. One "natural" extension that I found was $(Z_3,Z_2)$. That is $0 \rightarrow Z_3\rightarrow (Z_3,Z_2)\rightarrow Z_2\rightarrow 0$. ...
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Prove that $M$ is a complex.

Let $f:(A,d) \rightarrow (A^{'},d^{'})$ be a chain map. For each $n$ define $$M_{n}=A_{n-1} \oplus A^{'}_n$$ and $\Delta_{n} :M_{n} \rightarrow M_{n-1}$ by $$\Delta_{n}:(a_{n-1},a_{n}^{'}) ...
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35 views

What is maximal possible order of an element in $S_{10}$ ? Why?

What is maximal possible order of an element in $S_{10}$? Why? Give an example of such an element. How many elements will there be in $S_{10}$ of that order?
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Splitting in Short exact sequence

I am trying to find whether $\{1\}\longrightarrow\mathbb{Z}\longrightarrow\mathbb{R}\longrightarrow\mathbb{R}/\mathbb{Z}\longrightarrow \{1\}$ splits. My conjecture is it is not as we cannot find a ...
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1answer
31 views

Localizing at maximal ideals and the product

Let $D$ be an integral domain, $M_{i}$, $i = 1,...,r$ be some of its mutually distinct maximal ideals, and $e_{i}$be positive integers for all $i$. Is it true in general that the extension of the ...
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22 views

the order of a quotient group element

I'm looking at the solution to this question and there's one part that doesn't make sense to me. Let $G$ be a group and $N$ a normal subgroup of $G$. Prove that the order of the element $gN$ in ...
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56 views

Prove that : A group $G$ of order $p^n$ has normal subgroup of order $p^k$ , for all $0≤k≤n$

Prove that : A group $G$ of order $p^n$ has normal subgroup of order $p^k$ , for all $0≤k≤n$. I already appreciate your hints/answers
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2answers
50 views

Prove a given group is free

From Hungerford's "Algebra": What type of tools does one have to tackle a problem like this? I seem at a loss at how to show a group is free at all. One can consider each group as the homomorphic ...
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1answer
42 views

Tensor analog of Matrix Product

Given two $n \times n$ matrices $A$ and $B$, we can form their matrix product in the usual way. Is there a similar product for tensors? E.g., if one is given two $n \times n \times n$ tensors ...
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Why the Endomorphism ring of module is a proper class of that of abelian group?

Wikipedia: In the category of R modules the endomorphism ring of an R-module M will only use the R module homomorphisms, which are typically a proper subset of the abelian group homomorphisms. I ...
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Hilbert's Basis Theorem - Clever Proof?

So I am studying commutative algebra at the moment and I have come across the proof of the Hilbert Basis Theorem (the proof I have is the same as the one in Reid's "Undergraduate Commutative ...
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duality for (co)homology of Lie algebras

Let $R$ be a commutative ring and $\mathfrak{g}$ a Lie $R$-algebra that has an $R$-module basis with $n$ elements. What is the relationship between $H_k(\mathfrak{g};R)$, $H_{n-k}(\mathfrak{g};R)$, ...
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47 views

Show that $G_{s}$ is a normal subgroup of $G$

Definition: $G_{s}:=\{g \in G: g.s=s\}$ My attempt is the following: We take $g \in G$, and we consider this two sets: $$gG_{s}:=\{gh:h\in G_{s} \}$$ $$G_{s}g:=\{hg :h\in G_{s}\}$$ and we will ...
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467 views

Why are there so many universal properties in math?

I don't really understand why there are so many universal properties in math or why they all need to be highlighted. For example, I'm studying some Algebra right now. I have found three universal ...
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1answer
39 views

Character table of the non-abelian group of order 21

I'm working my way through the first Chapter of Fulton and Harris' Representation Theory and I'm trying exercise 3.26: There is a unique nonabelian group $G$ of order 21, which can be realized as ...
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22 views

Prove that a $\kappa : G/G_{s} \to G.s$ is a bijection

I have to prove that given an action this function $\kappa : G/G_{s} \to G.s$ is a bijection. $$ G/G_{s} \to G.s$$ $$gG_{s} \to g.s$$ Where $G$ is a group and: $G_{s}:=\{g \in G : g.s=s\}$(Isotropy ...
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22 views

How do you distinguish rings with unity?

Recently, I have changed my vocabulary in abstract algebra. If one doesn't require "ring" to have a unity, then one can say that "ring with unity $1\neq 0$" However, I changed my vocabulary to call ...
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a question about covariant and countravariant exact functor .

Let $T:‎_{R}‎\mathfrak{M}‎‎ \rightarrow ‎_{R}\mathfrak{M}‎$ be an exact (covariant) functor. For each $n \in \mathbb{Z}$ and every complex $A$ of R-modules, prove that $H_{n}(TA) \cong TH_{n}(A) $. ...
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Automorphism iff G is abelian

Let $G$ be a group. Prove the mapping $\alpha(g)=g^{-1}\forall g \in G$ is an automorphism iff $G$ is abelian. Proof (forwards): Assume $G$ is an automorphism. Show $ab=ba$. How would I even go about ...
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67 views

How many distinct subgroups of order 10 are there in a non-cyclic abelian group of order 20?

We are currently working with free abelian groups and finitely generated groups. The homework problem asks us to find the number of distinct subgroups of order 10 in a non-cyclic abelian group of ...
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128 views

Vector space as simple $K[x]$-module

I am trying to solve the problem: Let $V$ be a vector space and $T$ a linear transformation $T:V \to V$. Let $(V,T)$ be a $K[x]$-module. Show that $(V,T)$ is simple if and only if $V$ is finite ...
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54 views

Double check $G\sim H$ iff $G≈H$

Let $S$ be the collection of all groups. Define a relation on $S$ by $G \sim H$ iff $G ≈ H$. Prove that this is an equivalence relation. So $S$ is partitioned into isomorphism classes. Proof: Let ...
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1answer
24 views

Is there some standard way to determine the basis of $K(a_0, a_1, \ldots, a_n)$ considered as a vector space over $K$?

Let $L \supset K$ be a field and consider $K(a_0, a_1, \ldots, a_n)$ a field extension of $K$ with $a_i \in L$. Is there some standard way to determine a basis of $K(a_0, a_1, \ldots, a_n)$ ...
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What is $R$-algebra?

(To be clear, I mean a ring by a ring with unity.) Here is a definition in Wikipedia: Let $R$ be a commutative ring. Let $(M,+,\cdot)$ be an $R$-module. Let $\ast$ be a binary operation ...
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A dimension relation on non-degenerate bilinear forms

Let $E$ be a finite dimensional $k$ -vector space and $F$ a subspace. Let $f:E\times E \to k$ be a bilinear form which is non-degenerate on its restriction to $F$. Is it true that ${\rm dim } F + {\rm ...
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Writing out an Alternating Group

I am trying to write out what is in $A_4$ and $A_6$, their general form, not the whole $n!/2$ cause that would be a lot. My main question is how do I do that. I know they are all the even permutations ...
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32 views

Prove there exists a unique polynomial

I'm having trouble proving the following lemma: Let $ p $ be a prime and $ f \in \mathbb{Z}_p [x_1, \dots, x_n] $. Prove there exists a unique polynomial $ f^* \in \mathbb{Z}_p [x_1, \dots, x_n] $ ...
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1answer
30 views

Using Kronecker's theorem to construct a field with four elements

Use Kronecker's theorem to construct a field with four elements by adjoining a suitable root of $x^4-x$ to $\mathbb Z/2\mathbb Z$. Definition: A polynomial $f(x)\in F[x]$ splits over $F$ if it is ...
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Are there terminologies distinguishing modules over ring and rng?

Let $R$ be rng. Let $M$ be a left $R$-module. Let's say, after some verification, one realized that $R$ has a unity and it doesn't satisfy $1_R \cdot x$ for all $x\in M$. Hence, one cannot call $M$ ...
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Abelian Group (Alternative Proof)

Is there an alternative method to prove $(ab)^{2}=a^{2}b^{2}$ for all elements $a,b \in G \implies$ $(ab)^{-1}=a^{-1}b^{-1}$ for all elements $a,b \in G$. then the one I give below? Let $G$ be ...
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34 views

Subgroups of $ \mathbb{Z}^n $

I'm having a difficulty proving the following: Let $ H \leq \mathbb{Z}^n $ be a subgroup. Prove that $ H \simeq \mathbb{Z}^k $, $ k \leq n $. Moreover, prove that if $ H = \{(c_1, \dots, c_n) \in ...
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39 views

Let $G$ be a group, $x∈G$, $ a,b∈\Bbb Z$ and $a⊥b$. If $x^a=x^b$, then $x=1$.

There is a missing step in this proof: http://math.stackexchange.com/a/106292/135812 Lemma Let $G$ be a group, $x\in G$, $a,b\in \mathbb Z$ and $a\perp b$. If $x^a = x^b$, then $x=1$. ...
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1answer
41 views

Prove that the order of an element a of $S_n$ is the least common multiple of the lengths

Prove that the order of an element a of $S_n$ is the least common multiple of the lengths of the cycles which are obtained when a is written as a product of disjoint cyclic permutations.
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Prove $\operatorname{stab}_G(g \cdot x) = g\operatorname{stab}_G(x)g^{-1}$

Suppose a group $G$ acts on a set $X$. The stabilizer in $G$ of $x \in X$ is $$ \operatorname{stab}_G(x) = \{a \in G : a \cdot x = x \} $$ For each $g \in G$, let $$ g\operatorname{stab}_G(x)g^{-1} = ...
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How do I prove the universal mapping property of the field of quotients?

Reference : http://www.math.niu.edu/~beachy/abstract_algebra/study_guide/54.html Theorem 5.4.6 Let $R$ be an integral domain and $Q(R)$ be the field of quotients of $R$. Let $F$ be a ...
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Let $G = \text {Gal}(L/K)$. How do I see that $\phi \in G$ is completely determined by $\phi\mathbb( \sqrt[4] {3})$ and $\phi(i)$?

Let $K = \mathbb Q$ and $L = \mathbb( \sqrt[4] {3}, i)$. I see that $L \supset K$ is a an Galois extension. Also, after computations I've $[L : K] = 8$ (degree of $L$ over $K$ considered as a vector ...
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68 views

Description of an abelian group

I'm again stuck in an algebra exercise. I'm not sure if I understand the problem right. Could it be that I have to show that $\mathbb{Z}[i]/\gamma$ can be expressed by a product of finite abelian ...
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51 views

$G$ finite group with $H \leq G$ with $G = \bigcup_{g \in G} gHg^{-1} \implies H=G$

Let $G$ be a finite group and let $H$ be a subgroup of $G$. If $G = \bigcup_{g \in G} gHg^{-1}$, then I have to show $H=G$. Let $N(H) =\left\{ g \in G \ \bigl| \ gHg^{-1}=H\right\}$. What I have ...
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81 views

Why does $M\otimes k(\mathfrak{m})=M_\mathfrak{m}/\mathfrak{m}M_\mathfrak{m}$? (From Matsumura, proof of Theorem 4.8.)

Matsumura's Commutative Ring Theory, proof of Theorem 4.8, page 27, says: Let $A$ be a ring, $M$ a finite $A$-module, and $\mathfrak{m}$ a maximal ideal. If ...
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40 views

Finding the Center Z(G) for the group G of all invertible Matrices in M2(R) under Multiplication

So i know that basically i need to find which matrices in M2(R) that commute with all other invertible matrices in M2(R) so Z(G) = { AX=XA for all X belongs to G } how can i find those A matrices ?
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Non-cyclic normal subgroup

Suppose $H$ is a cyclic normal subgroup of $G$, then it is true that every subgroup of $H$ is normal in $G$. However, it is not true if $H$ is not cyclic. Is there any counterexample?
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Linear independence and grammar

Let $A$ be a commutative ring. Normally, the linear independence is a property of a subset or a family elements of an $A$-module. But one often sees statements like "$v$ and $w$ are linearly ...
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41 views

$M$ is a simple module if and only if $M \cong R/I$ for some $I$ maximal ideal in $R$.

I am trying to show the following statement (taken from Rotman's Advanced Modern Algebra): Let $M$ be an $R$-module. Then $M$ is a simple module if and only if $M \cong R/I$ for some $I$ maximal ...
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36 views

Every element in a ring different from 0 is invertible [closed]

True or false.Every element r in a ring R, different from 0 is invertible. It seems like it is true but how to go about proof if we consider Matrix then it is invertible if determinant is not 0 .
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Proving that a quotient space is formed from a vector space with W-affine subspaces.

I have been given a 2-part question which first states given a vector space (V,K) and W$\subseteq$V is a subspace. that a W-affine subspace S$\subseteq$V is one in which s,s' $\in$ S, s-s' $\in$ W and ...