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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Suppose two transitive G-sets X and Y are isomorphic as G-sets. Show that the two corresponding actions have the same kernel. [on hold]

Suppose two transitive G-sets X and Y are isomorphic as G-sets. Show that the two corresponding actions have the same kernel. Show by example that two transitive G-sets with actions having the same ...
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Identity permutation

I was reading a proof and at one point the proof said that the identity permutation is the only permutation that fixes every element. Is the identity permutation the only permutation that fixes every ...
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different factors of irreducible polynomials over a Galois extension does not share roots

Let $F$ be a field and $E/F$ a galois extension. Let $f\in F[x]$ be an irreducible polynomial. I'm trying to prove that all the irreducible factors of $f(x)$ over $E[x]$ have the same degree. If ...
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Homomorphism from a subgroup to a group is injective.

I'm reading a proof and I don't quite understand one step of the proof. We want to deduce that if G acts transitively on A then $ \bigcap_{\sigma \in G} \sigma G_{a} \sigma^{-1} = 1$. (Where $G_{a}$ ...
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2answers
49 views

Intuitive Understanding of the First Isomorphism Theorem

I've been reading some things about an intuitive understanding of the first isomorphism theorem, and there's just one more part that I do not understand. Specifically, the first isomorphism theorem ...
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In $S_3$, determine the set $T=\{ x\in S_3 | x^2=e\}$. Is $T$ a subgroup of $S_3$?

Here's my solution: Is it right or wrong? $S_3=\{ \begin{cases} 1\mapsto1 \\ 2\mapsto 2 \\ 3\mapsto 3 \end{cases}, \begin{cases} 1\mapsto 2 \\ 2\mapsto 1 \\ 3\mapsto 3\end{cases}, \begin{cases} ...
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1answer
11 views

Can the induced function of non-zero $f \in R[X]$ be zero, when $R$ is an infinite non-integral domain?

Let $R$ be an infinite commutative ring with $1$ which is not an integral domain. Is it possible to have a non-zero $f\in R[X]$ such that the induced map $\bar{f}: R \to R$ is zero? Please give ...
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1answer
10 views

Order of Elements of Q and Infinite Cyclic Groups in General

In my notes for class it says that elements of the additive group Q have either order 1 or infinity. The former part of this is obviously true (since id element will have order of 1). But why do all ...
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1answer
21 views

Show that every finite group G has a subgroup of order pq, for any choice of p and q such that p is prime, pq||G| and q||Z(G)| [on hold]

Show that every fi nite group G has a subgroup of order pq, for any choice of p and q such that p is prime, pq||G| and q||Z(G)|
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Presentation of a module by generators and relations

Let $R:=\mathbb C[T]$. Match the $R$-module with the presentation by generators and relations. $\bullet$$R$-modules: $M:=\mathbb C[T,T^{-1}]$ (Laurent Polynomials)$\qquad$$N:=\mathbb ...
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Image of Homomorphisms for Cyclic Groups

I'm reading an example for abstract algebra and I would like to clarify some terms. The example is showing the number of homomorphisms from $\mathbb Z $ to $S_3$. Take a homomorphism $f$ from ...
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57 views

xy+x+y=0 What is the inverse Element?

I have a task to show that $G:= \mathbb Q\setminus \{-1\}$ is a group to some binary operation * (x is the "times" symbol) with $A*B:=A\times B + A + B$ I could show that it's associative, that ...
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1answer
72 views

Describing all the homormorphisms from $\Bbb Z_{10}$ to $\Bbb Z_{15}$

I'm working on a problem that asks me to show all the homomorphisms from $\Bbb Z_{10}$ to $\Bbb Z_{15}$. So far, my attempt is as follows: Since $\Bbb Z_{10}$ and $\Bbb Z_{15}$ are both cyclic, I ...
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46 views

Centralizer of $A_m$ in $A_n$

Problem: I want to calculate centralizer of $A_m$ in $A_n$. As far as I know, $A_{i},\ i\geq 5$ is simple. So if $m\geq 5$, then $$ C := \{ a\in A_n|\ ax=xa\ \forall x\in A_m\} = A_{n-m} $$ ...
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1answer
16 views

Order of an Element Under a Homomorphism

I was reading an example about describing homomorphisms and I'm having a bit of trouble with this one. Show there is no group homomorphism $f : \mathbb{Z}_{10} \to \mathbb{Z}_{25}$ such that $f(1) = ...
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1answer
22 views

Polynomial in two variables

In the ring of polynomials of two variables $\mathbb{C}[x,y]$, the polynomial $f(x,y)=x^2+y^2+1$ is irreducible ?
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$H\trianglelefteq A$ and $K\trianglelefteq B\Rightarrow HK\trianglelefteq AB$?

Let $G$ be a group and $A, B\leq G$. Suppose $H\trianglelefteq A$ and $K\trianglelefteq B$. Is it true that $HK\trianglelefteq AB$? Notation: $\leq$ means subgroup and $\trianglelefteq$ means normal ...
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non-zero elements in $\mathbb Z_3[i]$ form an abelian group

How shall I show that all non-zero element of $\mathbb Z_3[i]$ form an abelian group of group of order $8$ under multiplication... Please any hint how shall I show this result?
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34 views

$\mathbb Z[\sqrt d]$ is not a field.

How shall I show that $\mathbb Z[\sqrt d]=\{a+b\sqrt d~~\big|a,b\in \mathbb Z\}$ is not a field? It is an integral domain, so the thing it lacks maybe is every element does not have a ...
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1answer
30 views

Equivalence of categories is an equivalence relation

Suppose that $B \xrightarrow{F}C$ and $C \xrightarrow{G}D$ are equivalences of categories. I want to show that $G \circ F$ is an equivalence. (This becomes easy if I use that a functor is an ...
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61 views

prove that $ab=1$ implies $ba=1$.

I have a doubt how to prove: If the product of any pair of non-zero elements of $R$ is non-zero , prove that $ab=1$ implies $ba=1$. how shall I make use of the fact : product of any pair of ...
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Canonical Structure to figure out proof strategy

Even though the integers form a Euclidean domain, most results about can be derived from the weaker fact that it is a PID (I do realize that establishing it forms a PID uses the fact that it is an ED, ...
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38 views

Multiplication of subgroups

Let $H$ and $K$ be subgroups of a finite group $G$. Define $HK = \{hk\mid h \in H, k \in K\}$ and $KH = \{kh\mid k \in K, h \in H\}$. a) Show that in general $HK \ne KH$. (For example, consider $G = ...
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43 views

Representation is reducible

Suppose $V$ is a representation of a finite group $G$ over a field $k$ of characteristic $0$, and suppose dim$V=3$ and $\wedge ^2V$ is reducible. Then $V$ is reducible. I was trying to do it by ...
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1answer
20 views

Center of a Group and automorphisms

The center of a group, G is defined as: Z = {a|ag = ga for all g ∈ G}. Let $\phi$ be an automorphism of a finite group G to G. Show that for any a ∈ Z then $\phi(a) $∈ Z. Conclude that $\phi(Z) = Z$. ...
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1answer
30 views

Zero divisors in $(\mathbb Z_n,+,*)$

How to understand this : An element $a$ in $(\mathbb Z_n,,+,*)$ is a zero divisor iff $a$ and $n$ aren't coprime... EDIT: Is it also true that an element a in $(\mathbb Z_n,,+,*)$ is a unit iff ...
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2answers
38 views

$\mathbb R\oplus \mathbb R$ is an Integral Domain or a Division Ring?

Can anyone help me with these two doubts of mine: Is the ring $\mathbb R\oplus \mathbb R$ an Integral Domain or a Division Ring? My notes state that the ring of Gaussian integers(i.e. $\mathbb ...
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1answer
37 views

What is a quick proof that $f \in \mathbb{C}[X_1,\dotsc,X_n]$ is determined by its induced function on $\mathbb{C}^n$?

For $f \in \mathbb{C}[X_1, \dotsc, X_n]$, we have the induced function $\bar{f}: \mathbb{C}^n \to \mathbb{C}$ given by evaluation. The association $f \mapsto \bar{f}$ is injective. Is there a quick ...
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107 views

Is product of prime ideals prime?

I'm trying to show that the product of ideals $(x_1, x_3)$ and $(x_2, x_4)$ in $\mathbb C[x_1, x_2, x_3, x_4]$ is a radical ideal, but no other way that I can think of works. So, is the product ...
3
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1answer
115 views

Questions related to maximal ideals

In my previous sessional exams, I was asked to prove these two: 1) Find a ring which doesn't have a maximal Ideal. 2) If a ring has unity, then it has a maximal Ideal. About the ...
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Find the kernel of the group homomorphism $G\to\text{Bij}(G/H),\;a\mapsto(b\mapsto abH)$

Let $G$ be a group and $H\subset G$ be a subgroup. Find the kernel of the group homomorphism $$G\to\text{Bij}(G/H),\;a\mapsto(b\mapsto abH)$$
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Suppose that a function $f: G \to H$ is an isomorphism? Must f inverse be an isomorphism as well?

Suppose that a function $f: G \to H$ is an isomorphism? Must f inverse be an isomorphism as well? Prove it if it is so, else, provide a counterexample. I think that it is, the following is my proof: ...
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1answer
35 views

Showing that an ideal is prime

I think that $k[x,y,z]/(z-1, x^2-y)$ can be identified as a subset of $k[x,y]$ with all polynomials whose $x$ terms are only degree one. Therefore I conclude that $k[x,y,z]/(z-1, x^2-y)$ is integral ...
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1answer
29 views

Does the dimension of the row space equal dimension of the column space for complex matrices?

In the case of real matrices, the dimension of the row space equals the dimension of the column space: $\text{dim } \text{span}(A) = \text{dim }\text{span}(A^T)$. But for complex matrices the ...
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1answer
33 views

no. of elements in $\mathbb Z[i]/\langle 3+i\rangle$. [duplicate]

$\mathbb Z[i]/\langle 3+i\rangle$ can be represented as :$\{a+3b+\langle 3+i\rangle\big|~~a,b\in \mathbb Z\}$ How shall I find the total no. of elements in $\mathbb Z[i]/\langle 3+i\rangle$.. ...
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50 views

What is so special about Klein 4-group?

This is my first course in abstract algebra and so far I am only learning about groups. So is there anyone who can explain to me why Klein 4-group is so special that it warrants a category of its own. ...
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1answer
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Universal property for vector spaces

QUESTION: I would like some hints for part b.I am unsure of how to show that $\widetilde{T}$ is surjective or injective for that matter. My construction for $\widetilde{T}$ is ...
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Follow-up regarding $f(x,y)\cdot f(y,z) = f(x,z)$

I posted this question about a year ago. And I started thinking about it again. The question as follows: Under what conditions does $f(x,y) \cdot f(y,z) = f(x,z)$ not imply that $f(x,y)$ can be ...
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4answers
49 views

Insights on coset in abstract algebra

I am taking course on abstract algebra. I found the concept of coset is very important. However, I have not found any insights on my textbook (and by myself). I believe there must be great example on ...
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1answer
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Show that the element $z=i \cos \frac{\pi}{3}+\sin \frac{\pi}{3} = i( \cos \frac{\pi}{3} - i \sin \frac{\pi}{3})$ belongs to $U_{12}$

Show that the element $\displaystyle z=i \cos \frac{\pi}{3}+\sin \frac{\pi}{3} = i( \cos \frac{\pi}{3} - i \sin \frac{\pi}{3})$ belongs to U12 What I don't understand: In what way $i \cos ...
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Lucas' Theorem for $p$-adic integers?

Does Lucas' Theorem hold for $p$-adic integers? More specifically, does it specifically hold for the case that, given a $p$-adic integer: $$x = x_0 + x_1p + x_2p^2 + \cdots = \sum_{i=0}^\infty ...
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1answer
31 views

Show that $a \star b=a \cdot b+a+b$ is binary operation for the group $\Bbb{ Q} - \{-1\}$

The group $\left(\Bbb{ Q} - \{-1\},\star\right)$ has as its underlying set the rational numbers different from $-1$ and the operation $\star$ is defined as $a \star b=a \cdot b+a+b$ where ...
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1answer
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Constructing a colimit

Consider a functor $F: \mathcal{I} \rightarrow \textrm{Sets}$ where $\mathcal{I}$ is small. Then the colimit of $F$ is given by $\amalg_{i \in \textrm{Ob}(\mathcal{I})}F(i)/{\sim}$ where $\sim$ is the ...
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1answer
26 views

Something similar to Euler's theorem

If $p$, $q$ are not equal primes. $n=pq$, $\varphi(n) = (p − 1)(q − 1)$, $d = \gcd(p − 1, q − 1)$. Is it true that for any $a$ such that $\gcd(a, n) = 1$ holds $a^{\frac{\varphi(n)}{d}} \equiv 1 ...
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1answer
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Is there a basis for the continuous functions space?

I've been searching all over the Internet for this but without finding a satisfying answer. This might be a dumb question, but I would like to know the answer anyway. Is there a set of continuous ...
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The diophantine equation $A^2+B^2=C^2$ for integer-valued polynomials

How can I find the solutions to this diophantine equation in $\Bbb{Z}[X]$: $$A^2+B^2=C^2 \, ?$$ Here $A$, $B$, $C$ are polynomials.
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42 views

Proving commutativity [duplicate]

Let $R$ be a ring in which $x^2=x$ for all $x\in R$ where $x^2$ of course denotes $x\cdot x$. a. prove that $x+x=0$, for all $x \in R$ b. prove that $R$ is commutative. I have done part a but how ...
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93 views

Is abstract algebra (mostly?) restricted to $2$-ary operators?

This may be due to my own pure ignorance but it's my experience that all abstract algebra I've been introduced to, both in actual courses and in self-studies only exclusively deals with algebraic ...
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2answers
57 views

Logic problem: Atiyah-Macdonald 1.11

Proposition 1.11 in Atiyah-Macdonald's "Introduction to commutative algebra" states the following: "Given an ideal $I$ in a ring $A$ and $p_1, \dots p_n$ prime ideals, then $I \subset \cup_i p_i$ ...
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Internal Direct Sum [on hold]

Let F be a field and V a vector space over F. Let $W_1$, $W_2$, U be proper subspaces of V. Assume that each $W_i$ is a complement of U, that is $W_i \oplus U=V$ is an internal direct sum. Do not ...