Tagged Questions

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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4
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0answers
15 views

Pronunciation of `Rng` - the non-unital Ring

I chuckled the first time I heard that a Ring without a multiplicative identity (Ring without the i) is called a ...
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5 views

Minimal polynomial in $T$-invariant subspace

I am stuck on the following problem. Problem: Let $V$ be a finite dimensional vector space over field $F$ and $T$ a linear transformation from $V$ to $V$. $W$ is an invariant subspace. Let $h_1$ be ...
1
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1answer
21 views

A simple question about free group

Fix $r\in \mathbb{N}$ and let $\mathbb{F}_{r}=\langle g_{1}, ...,g_{r}\rangle$ be the rank-r free group. I have asked a question several days ago: Is $\mathbb{F}_{2}$ a subgroup of $\mathbb{F}_{3}$? ...
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0answers
11 views

Suppose $H:= \{\sigma \in G| \sigma(1) = 1\}$, if for any $j \in \{1,2,…,n\}$ $t_j\in G$ such that $t_j(1) = j$. Show that $|G| = n|H|.$

Let G be a subgroup of the symmetric group $S_n$ in n letters. Consider the following subset of G: $$H:= \{\sigma \in G| \sigma(1) = 1\}$$ Suppose that G acts on the set $\{1,2,...,n\}$ transitively ...
0
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2answers
22 views

elementary algebra question On the generator of a group

(Def): Let $G$ be a group and $X \subset G$. Let $\{ H_{\alpha} \}_{\alpha \in \Gamma} $ be a collection of all subgroups of $G$ which contain $X$. Then $\bigcap_{\alpha \in \Gamma} H_{\alpha} $ is ...
0
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1answer
19 views

Prove $\text{rad}(I)/I$ is isomorphic to $\mathfrak{N}(R/I)$

I want to know if this is the correct way to do it. Define $\varphi:\text{rad}(I) \longrightarrow \mathfrak{N}(R/I)$ by $\varphi(r)= r^n+I$,then ker$\varphi = I$, so therefore by the 1st isomorphism ...
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0answers
11 views

Multiplication in symmetric product space

STATEMENT: Let $V=\mathbb{R}^2$.Take $Y:=\left\{x\cdot y: x,y\in S_2(V)\right\}$ where $S_2(V)$ is the symmetric product of $V$. QUESTION: What is multiplication in the symmetric product space?
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10 views

$F/E$ a finite Galois extension, the integral closure of $E[X]$ in $F(X)$

If $F/E$ is a finite Galois extension, then one can show that $F(X)/E(X)$ is also a finite Galois extension of the same degree (a basis for $F/E$ is also a basis for $F(X)/E(X)$). Since $E[X]$ is a ...
1
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1answer
27 views

Question about the conjugation of an element in a group

Let $(a_1,a_2,a_3)$ be a 3-cycle in the alternating group $A_4$ in four letters. Find $g \in A_4$ such that $$g(a_1,a_2,a_3)g^{-1}=(a_2,a_1,a_4) = (a_1,a_2,a_4)^{-1}$$ Why do we need the last ...
2
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0answers
35 views

Splitting field of $1 + x^3 + x^6 + x^9$

I'm studying for a qualifying exam and wanted to know if my reasoning on this problem was correct: Let $L$ be the splitting field of $f(x) = 1 + x^3 + x^6 + x^9$ over $\mathbb{Q}$. Find the Galois ...
2
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0answers
15 views

Is $T:= \{g \in A_4|g^2 =(1)\}$ a subgroup of $A_4$?

Consider the subset $$T:= \{g \in A_4|g^2 =(1)\}$$ of the alternating group $A_4$ in four letters. Is T a subgroup of $A_4$? My Proof: Yes. If I am not wrong T is the Klein 4-group since only ...
1
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1answer
65 views

Suppose that half of the elements of G have order 2 and the other half forma a subgroup H of oder n. Prove that H is an abelian subgroup of G.

Let $n>1$ be a positive integer. Let $G$ be a group of order $2n$. Suppose that half of the elements of G have order 2 and the other half forma a subgroup H of oder n. Prove that H is an abelian ...
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0answers
26 views

What are some of the effective methods and theorems to prove that a group or subgroup is abelian? [on hold]

What are some of the effective methods and theorems to prove that a group or subgroup is abelian? Can someone give me a list of them based on your experience. Thanks. Now I have always been trying to ...
-2
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1answer
32 views

Trick with numbers, sums, cubes, squares? [on hold]

Let $(X_1, X_2, \ldots , X_n) = 10$ be a sum of positive numbers where $(X_1^2, X_2^2, \ldots , X_n^2) \geq 20$ is a sum of their squares. Prove that $(X_1^3, X_2^3, \ldots , X_n^3) \geq 40$.
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3answers
36 views

Trying to calculate the quotient group $\mathbb{Z}\times\mathbb{Z}/\langle (1,1),(1,-1)\rangle$

Let $G$ be the group $\mathbb{Z} \times \mathbb{Z}$ and $H$ be the subgroup of $G$ generated by $(1,1)$ and $(-1,1)$. I am trying to calculate the quotient group $G/H$.
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0answers
31 views

The number of solutions of $x^n = e$ in a finite group is a multiple of n, whenever n divides the group order.

Prove that in a finite group G the number of solutions of the equation $x^n = e$ is a multiple of n, whenever n divides the order of the group. I feel there is a very simple answer to this question, ...
0
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0answers
27 views

How is number of conjugacy class related to the order of a group?

Let $c(G)$ denote the number of conjugacy classes of a group $G$. I have to show that $$\lim_{|G|\to \infty} c(G)=\infty.$$ That is, I have to show that $\exists $ a function $f:\mathbb{N} \to ...
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1answer
37 views

Prove every prime ideal of a ring is a radical ideal.

this is my attempt: Since $R$ is commutative, we let $I$ to be a prime ideal of $R$, the for $a,b\in R$,then the product $ab$ we must have that $a\in I$ or $b \in I$, by definition of a prime ideal. ...
2
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1answer
36 views
1
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1answer
47 views

Prove either $G=ST$ or |$G|\geq|S|+|T|$

Let G be a finite group, and let S and T be (not necessarily distinct) nonempty subsets. prove that either $G=ST$ or |$G|\geq|S|+|T|$ That's my thougt, I am thinking suppose $G$ does not equal to ...
0
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1answer
14 views

Notation question regarding field extensions (What does $K^2 \subseteq k$ mean)

recently I am reading a paper on pfister forms in characteristic 2 and stumbled across a notation I do not know. It can be found here Suppose $k$ is an arbitrary field of characteristic 2. Let ...
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2answers
33 views

In ring theory, what does $R^{2} \neq \{0\}$ mean?

I'm working on an exercise of Malik's Fundamentals of Abstract Algebra, namely: "Let $R$ be a ring such that $R^{2} \neq \{0\}$. Prove that $R$ is a division ring if and only if $R$ has no nontrivial ...
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0answers
30 views

$(a,b)[a,b]=ab$ in non factorial monoids

Do you know of a proof of $[a,b](a,b)=ab$ in $\mathbb Z$ that doesn't use prime factorization? To be more precise let's strip all unnecessary properties and leave only the bare bones of divisibility: ...
0
votes
1answer
14 views

Two quotient morphisms and universal property

I am reading some notes on group theory and I am having some doubts related to the following: Let $S \lhd G$ and let $\rho:G \to Q, \space \rho': G \to Q'$ be two quotients of $G$ by $S$. Then, by ...
0
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1answer
25 views

Using induction to prove an abstract problem

prove by induction that the order of a permutation $S_n$ is $n!$ This is what we have so far. Proof: Let $p(n)$ be a proposition. $P(n): S_n = n!$ base case: $n=1$ then $(1) = 1! = 1$ Inductive ...
0
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1answer
26 views

Factor Group, Isomorphism

If group $A=S_3⊕\mathbb{Z}_4$ and subgroup $B=\langle (132),2\rangle$, find a group the factor group $A/B$ is isomorphic to and construct the group table for $A/B$. I'm really not sure what to do with ...
0
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1answer
24 views

Induced invariant linear map in the dual space

This is the problem that I am stuck on. Problem: Let $V$ be a finite dimensional vector space and $T: V\rightarrow V$ be a linear transformation. Suppose ...
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0answers
28 views

Finding the normal subgroups in a semidirect product

Let the group $A=\cdots\times\mathbb{Z}_{-1}\times\mathbb{Z}_{0}\times\mathbb{Z}_{1}\times\cdots$ with $\mathbb{Z}_{i}=\left\langle a_{i}\right\rangle $ and $\alpha:a_{i}\rightarrow a_{i+1}$ an ...
0
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1answer
19 views

Left & Right Cosets of a Kernel

If we let $\phi:\mathbb R^*\to \mathbb R^*$ under multiplication be given by $\phi(x)=$| $x$ |. What are the left and right cosets of the kernel? Any assistance would be appreciated.
3
votes
1answer
56 views

If the commutator of a finite group has order $2$, then the order of the group is divisible by $8$

Prove that if $|G| < \infty$ and $|G'| = 2$ then $|G|$ is divisible by $8$. Thoughts. $A \simeq G / G'$ is abelian and $G' \simeq \mathbb{Z}_2$. Since $G' \subset G$ then at least $|G| \vdots 2$. ...
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0answers
32 views

Is quotient under $S_4$ action on “cube” representation a flat morphism?

Consider a three-dimensional irreducible representation $V$ of $S_4$, corresponding to symmetries of cube. Let $p$ be canonical projection $p: V \rightarrow V/S_4$. My question: is $p$ flat? I want ...
1
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1answer
22 views

Unique image of torsion groups in the circle group.

Let $p$ be a prime. For any one-to-one homomorphisms $f,g:\Bbb Z_{p^\infty}\to \Bbb T$, we have $f[\Bbb Z_{p^\infty}]=g[\Bbb Z_{p^\infty}]$, where $\Bbb T $ is the circle group. Is this correct ...
3
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1answer
55 views

Is $\Bbb{Q}[x,y]/(x)=\Bbb{Q}[y]$?

Is $\Bbb{Q}[x,y]/(x)=\Bbb{Q}[y]$ for all practical purposes? I used to think so, but my friend says that there are subtle differences between the two. I fail to grasp them.
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0answers
17 views

On Galois closure

I'm working on this problem in Hungerford: For $\sigma \in Aut_F \bar{F}$, show that every finite extension of $K$, the fixed field of $\sigma$, is cyclic. For a finite extension $L$ of $K$, let $M$ ...
3
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0answers
22 views

Can we show it without involving that $V=V^{**}$ are canonically isomorph?

My text proves the following Theorem. Let $V$ be a vector space over $F$ and $B=\{ v_1, \ldots , v_n \}$ a basis of $V$. Then there is exactly one basis $B^*=\{ f_1, \ldots , f_n \}$ of $V^*$ with ...
4
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0answers
32 views

homomorphism inducing monomorphism on some quotient group

Let $f:G\rightarrow H$ be a group homomorphism such that $f_* :G_{ab}\rightarrow H_{ab}$ is an isomorphism and that $f_* : H_2(G)\rightarrow H_2(H)$ is an epimorphism. Question is to prove that this ...
0
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1answer
26 views

algebra of polynomials of non-commuting variables

In Hatcher's book algebraic topology page 496-497: the Steenrod algebra $\mathcal{A}_2$ is the algebra over $\mathbb{Z}_2$ formed by polynomials of non-commuting variables $Sq^1,Sq^2,\cdots$ modulo ...
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0answers
30 views

Find a complex number for the splitting field

Let $E$ the splitting field of $x^3-2 \in \mathbb{Q}[x]$. Applying the algorithm of the proof of the Primitive element theorem, find a complex $c$ with $E=\mathbb{Q}(c)$. $$$$ I have done the ...
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0answers
11 views

find $f(x) \in \Bbb Z[x]$ s.t $f(x)-\frac {p(x)}{x^k} \in P=\{\frac {(x^2+1)q(x)}{x^k}| k\geq 0, q(x) \in \Bbb Z[x]\}$ where $p(x)\in \Bbb Z[x]$.

Given $\displaystyle\frac {p(x)}{x^k}$, find $f(x) \in \Bbb Z[x]$ s.t. $f(x)-\displaystyle\frac {p(x)}{x^k} \in P=\{\frac {(x^2+1)q(x)}{x^k}\mid k\geq 0, q(x) \in \Bbb Z[x]\}$ where $p(x)\in \Bbb ...
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0answers
18 views

Center of a ring projective?

If $R$ is a ring and $Z(R)$ denotes $R$'s centre, then when is $R$ projective as an $Z(R)$-module?
3
votes
1answer
34 views

Universal property of the algebraic closure of a field

At page 4 of Strom's "Modern Classical Homotopy Theory" there is a universal formulation of the algebraic closure of a field. You can read it here from google books. Exercise 1.2a is then to convince ...
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2answers
32 views

Set notation query

What do square brackets mean next to sets? Like $\mathbb{Z}[\sqrt{-5}]$, for instance. I'm starting to assume it depends on context because google is of no use.
0
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1answer
28 views

Global dimension of the center

Let $R$ be a ring. Must the global dimension of the centre $Z(R)$ of the ring $R$ always be atmost that of $R$ itself? I mean is it generally true that: $D(Z(R)) \leq D(R)$ (where D is the global ...
6
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1answer
115 views

can anyone give a proof by definition: $11$ is prime in $ \mathbb{Z}[\sqrt{-5}] $

what i did is: assume $\alpha \notin (11),\beta\notin (11), \alpha\beta \in (11)\Rightarrow\exists \gamma, s.t.$ $ \alpha\beta = 11 \gamma$, $\Rightarrow N(\alpha)N(\beta) = 11^2N(\gamma) $ then ...
0
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0answers
49 views

Irreducibility over Q doesn't imply irreducibility over R

I want a counterexample of polynomial that is irreducible over $\mathbb Q$ but not irreducible over $\mathbb R$ (i.e not maximal over $\mathbb R$).
14
votes
3answers
149 views

$|G| + \frac{|G|}{\left|\langle a\rangle\right|} + \frac{|G|}{\left|\langle b\rangle\right|} + \frac{|G|}{\left|\langle ab\rangle\right|}$

Show that for every finite group $G$ and for every elements $a, b \in G$ the following expression $$ |G| + \frac{|G|}{\left|\langle a\rangle\right|} + \frac{|G|}{\left|\langle b\rangle\right|} + ...
1
vote
2answers
40 views

Show that $G' = \bigcap_{C\subseteq N \triangleleft G} N$

I am asked to show that $G' = \bigcap_{C\subseteq N \triangleleft G} N$, where $G'$ is the commutator subgroup of $G$, and $C :=\{aba^{-1}b^{-1}\mid a,b\in G\}$. Showing $\bigcap_{C\subseteq N ...
0
votes
1answer
16 views

degree of extension of rationals adjoin infinitely many roots

Consider a radical extension of $\mathbb{Q}$ in which infinitely many numbers are adjoined. Each of these numbers is the $n^{th}$ root of some integer, where $n$ can vary. Can any information be ...
1
vote
1answer
25 views

What are some examples of these kinds of commutative semirings?

What are some examples of commutative semirings such that the following hold? Multiplication is idempotent i.e. we have $xx=x$ for all elements $x$. Addition is not idempotent i.e. there is at least ...
-2
votes
0answers
22 views

What do we mean by a group geometrically? [on hold]

What do we mean by a group geometrically? Can we study algebra geometrically? If so, give some articles or books regarding this..