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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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 ...
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51 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 ...
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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 ...
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1answer
73 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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What are some of the effective methods and theorems to prove that a group or subgroup is abelian? [closed]

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 ...
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3answers
61 views

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

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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48 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, ...
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2answers
113 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 $$\lim_{n \to \infty} \inf _{|G|=n}c(G)=\infty.$$ That is, I have to show that $\exists $ a function ...
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45 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. ...
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45 views

If $p_1, p_2,…,p_s$ are distinct primes, show that an abelian group of order $p_1p_2\cdots p_s$ must be cyclic

Can anyone give me some clues on this? Specific steps needed! Thank you!
2
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1answer
61 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 ...
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1answer
15 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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37 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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2answers
89 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: ...
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1answer
15 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 ...
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113 views

On polynomials over finite fields?

Pick prime $q\in\Bbb Z$ such that $q>B^{3t}>(mB)^{2t}$. Suppose I have a multilinear polynomial $g(x)\in \Bbb Z[x_1,\dots,x_n]$ of degree $t$ with $m^t$ non-zero coefficients that bound by ...
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1answer
30 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 ...
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34 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 ...
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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 ...
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53 views
+50

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.
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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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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 ...
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1answer
25 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
56 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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1answer
51 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 $M$ ...
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25 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 ...
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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 ...
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1answer
41 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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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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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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21 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?
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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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35 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.
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1answer
29 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 ...
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1answer
122 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 ...
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0answers
52 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
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3answers
175 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|} + ...
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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 ...
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1answer
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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 ...
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1answer
26 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 ...
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What do we mean by a group geometrically? [closed]

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

Cyclotomic polynomial identity [duplicate]

STATEMENT Show that $$\Phi_n(X)=\prod_{d\mid n}(X^{n/d}-1)^{\mu(d)}$$ where $\Phi_n(X)$ is the n-th cyclotomic polynomial. QUESTION Could anyone offer any help on how to show this result.
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algebraic extension by polynomials roots

Using solubility by radicals, Is possible to prove that $\mathbb{Q}(R_{\xi}) \neq \overline{\mathbb{Q}}$ where $R_{\xi}$ is a set of the unit root. I'm trying to generalize this, i.e., show that if ...
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2answers
32 views

$\mathbb{Q} \simeq \mathbb{Q}^*_+$ isomorphism [duplicate]

Is it true that $(\mathbb{Q},+) \simeq (\mathbb{Q}^*_+, \times)$? If yes then is there any constructive isomorphism?
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If $M$ is a $R$-module with $R=R_1\times \cdots \times R_n$ then $M\simeq M_1\times \cdots \times M_n$ where $M_i$ is a $R_i$-module?

Let $R_1, \ldots, R_n$ be rings and consider the ring $R:=R_1\times \cdots\times R_n$. How can I show every $R$-module $M$ is isomorphic to a product $M_1\times \cdots\times M_n$ where each $M_i$ is a ...
1
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1answer
36 views

How does one actually take the dual of a chain complex?

I know the following about the chain complex used for computing the homology groups of the torus $S^1 \times S^1$: The complex is $0 \to^{\delta_3} \mathbb{Z}[U] \oplus \mathbb{Z}[L] \to^{\delta_2} ...
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1answer
38 views

Prove that any subfield of $\Bbb R$ contains $\Bbb Q$

Prove that any subfield of $\Bbb R$ must contain $\Bbb Q$. Now for any subfield $F$ of $\Bbb R$, $1\in F$ so, $\Bbb Z \subset F \Rightarrow \Bbb Q \subseteq F$. Have I done it correctly?
2
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1answer
52 views

Determining Lie algebras from commutative diagrams of exact sequences

Suppose that we have the following commutative diagram of graded Lie algebras $$\begin{array} A 0& {\longrightarrow} & C_n & {\longrightarrow} & A_{n+1} &{\longrightarrow} & ...
3
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1answer
65 views

Example of $aH \subsetneq Ha$

Problem. Is there an example of a group $G$, a subgroup $H$ and an element $a \in G$ such that $|G : H| < \infty$ and $aH \subsetneq Ha$?
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1answer
43 views

Explanation of Proof that field sum of more than 2 elements is 0. [duplicate]

"Suppose the field $F$ is finite. If $f\colon F\to F$ is any bijection, then we can conclude that $\sum_{x\in F}x=\sum_{x\in F}f(x)$. Let $\alpha\in F$ such that $\alpha\ne 0$. Then $x\mapsto \alpha ...