Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, among other topics.

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Classifying groups of order $6$ using semidirect products

Let G be a group of order 6. I am able to do the exercise without semidirect products($G \cong Z_6 $ or $S_3$) but I don't know how to use semidirect products to do this. By Sylow's theorem, there ...
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1answer
25 views

Why do polynomials and integers both have a long division algorithm?

The grade-school long division algorithm and the polynomial long division algorithm are identical, if I'm not mistaken. Why is this the case? Are the two algebraic structures identical in some sense? ...
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14 views

Spliting subspaces and fields

I'm sure that the following is true, but I can't prove it. Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a tower of finite fields and $A = \{\theta\in K: ...
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$2\otimes 1$ is non-zero in $2\mathbb Z\otimes_{\mathbb Z} \mathbb Z/2\mathbb Z$.

I had the following doubt: Show that the element $2\otimes 1$ is $0$ in $\mathbb Z\otimes_{\mathbb Z} \mathbb Z/2\mathbb Z$ but not a zero in $2\mathbb Z\otimes_{\mathbb Z} \mathbb Z/2\mathbb Z$. ...
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3answers
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Prove that the four roots of unity form an abelian multiplicative group

My question is: Let $i = \sqrt{-1}$. Prove that the four roots of unity $\{1, -1, i, -i\}$ form an abelian multiplicative group. I know that abelian group is a group with commutative property. ...
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1answer
29 views

Order of a subgroup generated by two permutations

Let $$\alpha=(1,3,12,7)(8,5,6,2,11)(4,9,10)$$ $$\beta = (1,5)(6,8,11)(12,3,2,7)(4,9,10).$$ How does one prove that a subgroup G that contains $\alpha$ and $\beta$ has order $o \ge 120$? ...
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29 views

Permutations of $S_n$ whose order divides a positive integer $m$

For which $n,m \in \mathbb{N}$ is $$K_m = \{\sigma \in S_{n}: \text{ord}(\sigma) \text{ divides } m \}$$ a subgroup of $S_{n}$? How does one approach such a problem?
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2answers
25 views

Adjoint pair of functors and cogenerator elements

Let $F:\mathcal{A} \rightarrow \mathcal{B}$ and $G:\mathcal{B} \rightarrow \mathcal{A}$ be additive functors between abelian categories, such that $(F,G)$ is an adjoint pair. If $B \in \mathcal{B}$ is ...
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2answers
38 views

Is $(\mathbb{Z}_4, +_4)$ isomorphic to $(\langle i\rangle, *)$

Now, I am not sure, but I think that the second group is cyclic, because of the way it's defined $(\langle i\rangle,*)$. $i$ is probably the generator of the group. But, how can I prove that ...
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1answer
24 views

A group $G$ is finitely generated iff if there is a surjective homomorphism $F(\{1,…,n \}) \to G$

This is taken for granted in Algebra: Chapter 0 by Paolo Aluffi. Here is a definition of subgroup generated by a subset from the book: Let $A \subseteq G$. We have a ujnique group homomorphism ...
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1answer
44 views

Find error in abstract algebra proof

I suspect that the proof below is flawed. I did not use the hypothesis "$\ker(h) \subseteq \ker(k)$" when proving sufficiency. Lemma. $ $ Let $G$, $H$, $K$ be groups, let $h : G \to H$ and $k : G ...
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1answer
29 views

Suppose that $f : C → C$ is an isometry such that $f(0) = 0$, $f(1) = 1$ and $f(i) = −i$. Prove that $f(z) = \bar z$ for all $z ∈ C$.

Suppose that $f : C → C$ is an isometry such that $f(0) = 0$, $f(1) = 1$ and $f(i) = −i$. Prove that $f(z) = \bar z$ for all $z ∈ C$. I already have a proof for this but I would like an explanation ...
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1answer
28 views

Describing the clopen sets of a profinite group

I've read somewhere that all clopen subsets of a profinite group $$G \simeq \varprojlim\left(G_i, f_{ij}:G_i \to G_j\right)_{i,j \in I}$$ are exactly the preimages of subsets of the $G_i$'s. It's easy ...
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2answers
16 views

Show that the set of all principal ideals is an equivalence class of the relation $\sim$

Let $A$ a integral domain and let $\mho(A)$ the set of all non-zero ideals. Show that the set of all principal ideals is an equivalence class of the relation $\sim$ that we can noted by $[A]$. ...
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Norm on “tower” of fields (the question comes from algebraic number theory)

Consider the field extensions $L\supset K'\supset K$, where both $L|K$ and $K'|K$ are finite and Galois. I want to prove that $$N_{L|K}(L^\ast)\subseteq N_{L|K'}(L^{\ast})$$ Maybe it is very easy, ...
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1answer
20 views

What are the $GL_n(F)$-orbits of a group action on the set of idempotent matrices?

Let $S= \{A \in M_{n \times n}(F):A^2=A\}$ (set of idempotent matrices). The general linear group $G=GL_n(F)$ acts on $S$ by $A.g=gAg^{-1}$ (conjugation). I'm having trouble visualizing the ...
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2answers
26 views

Find $\phi, \psi: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, each nilpotent of order 2, whose composition is idempotent

$\phi: V \rightarrow V$ is nilpotent of order 2 if $\phi \phi$ is the zero endomorphism. Now composition of two such endomorphisms need not be nilpotent of order 2. Find $\phi, \psi: \mathbb{R}^2 ...
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Isomorphism from $\mathbb{Q}(\sqrt{2})$ to $\mathbb{Q}[x]/\langle x^2 - 2\rangle$ [on hold]

I am just now beginning my first course in Fields. Sometimes I learn best by just being absolutely certain of some basic facts. This is why I like to ask simple True/False questions that I think are ...
2
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1answer
31 views

Subrings of the product of rings of algebraic integers

Let $K,L$ be a number fields and $\mathcal{O}_K,\mathcal{O}_L$ their rings of algebraic integers, and $n_K=[K:\mathbb Q]$, $n_L=[L:\mathbb Q]$. Suppose that $R$ is a subring of ...
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1answer
31 views

I do not understand the definition of a “K-automorphism”

I am reading Ian Stewart's Galois Theory and frankly, the following definition is puzzling me Let $L:K$ be a field extension, so that $K$ is a subfield of the subfield $L$ of $\mathbb{C}$. A ...
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1answer
39 views

Show that $\mid$G$\mid$ is prime

Suppose Group G has precisely two subgroups. Show that $\mid$G$\mid$ is prime. $\mid$G$\mid$ denoted to the order of the group G. If I let $g\leq$ G and $k\leq$ G. I know that $G^p$= e where e is ...
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1answer
29 views

Prove closed disc $D^n$ is homeomorphic to the cone $CS^{n-1}$

I need to find a continuous surjective map from $D^n$ to $CS^{n-1}$. For 2 dimensions, we can use $$f: S^1 \times I /S^{1} \times \{1\} \rightarrow D^2$$ with $f(\theta,t) = (1-t)e^{i \theta}$ ...
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1answer
26 views

If $G/K$ is isomorphic to $H$ then is $G$ isomorphic to $KH$?

I don't know if the question is trivial but I would really appreciate if someone could prove/disprove this question. Thank you in advance Edit: I meant this: Suppose that there is homomorphism $f$ ...
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47 views

What kind of algebraic structure is $\left( \mathbb{R}_{\geq 0},+,\cdot \right)$?

Let $\left( \mathbb{R}_{\geq 0},+,\cdot \right)$ denote the non-negative real numbers with usual addition and usual multiplication. Obviously, this is not a field, because $0$ is the only additively ...
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1answer
11 views

Calculating $| \langle \cup_{i=1}^n P_i \rangle |$ where $P_i$ are Sylow subgroups of G

I'm trying to prove: Let $\lbrace P_i: i \in I \rbrace$ be a set of Sylow subgroups of a finite group G, one for each prime divisor of $|G|$. Then $\langle \cup_{i \in I} P_i \rangle = G$. (From ...
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1answer
36 views

Define $\phi:\mathbb{R}^3 \rightarrow \mathbb{R}$ by $\phi(e_1) = 1$, $\phi(e_2) = 2$, $\phi(e_3)=-1$. Determine ker$\phi$ and im$\phi$

Let {$e_1,e_2,e_3$} be the standard basis for $\mathbb{R}^3$ and define $\phi:\mathbb{R}^3 \rightarrow \mathbb{R}$ by $\phi(e_1) = 1$, $\phi(e_2) = 2$, $\phi(e_3)=-1$. Determine the subspaces ...
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2answers
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CG-modules: what does this notation mean?

I am trying to solve a question, but I do not know what the notation used means. If anyone could help me out that'd be great! I don't need help doing the proof, just what the notation means would be ...
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1answer
44 views

$\mathbb C$-dimension of vector space $\mathbb C\otimes_{\mathbb R}\mathbb C$

Let $\mathbb R$ be the field of real numbers, $\mathbb C$ be the field of complex numbers. Consider $\mathbb C\otimes_{\mathbb R}\mathbb C$ as a $\mathbb C$-vector space via $a(b\otimes c) := ab ...
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Prove that $u$ is a trivial unit

Let $G$ be a finite group. I want to prove that if $u\in \mathbb{Z}G$ be a torsion unit (i.e. for some $n$, $u^n=1$) of integral group ring $\mathbb{Z}G$ such that $u$ normalizes $G$, then $u$ is a ...
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2answers
32 views

Proving that rotation is an isometry in the complex plane

Consider the rotation $ρ_θ : \Bbb C → \Bbb C$ about the origin with angle $θ$ in counterclockwise direction; this can be described by the map $ρ_θ(z) = e ^{iθ} z$. Prove that $ρ_θ$ is an isometry of ...
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1answer
36 views

Show that $P_n$ is an $(n+1)$-dimensional subspace of the vector space of all real polynomials [duplicate]

Show that $P_n$ = {polynomials with real coefficients of degree $\leq$ n} is an ($n+1$)-dimensional subspace of the infinite-dimensional vector space of all real polynomials I know that $P_n$ is ...
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1answer
16 views

Showing that a function is an isometry of the complex plane and showing that a composition of functions in the complex plane is a translation

a). Let $a ∈ \Bbb{C}$ be fixed. Show that the map $T_a : \Bbb C → \Bbb C$ given by $T_a(z) = z + a$ is an isometry of $\Bbb C$. This is a translation of the complex plane $\Bbb C$. For this first ...
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1answer
9 views

Roots of the reduction of $x^p - (p-1)x^{p-1}-x+(p-1)^p \in \mathbb{Z}[x]$ modulo $p$

Let $p$ be a prime. Let $$g(x) = x^p - (p-1)x^{p-1}-x+(p-1)^p \in \mathbb{Z}[x].$$ How does one prove that the reduction of $g(x)$ modulo $p$ has exactly one root of multiplicity $2$ and the other ...
3
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2answers
61 views

In group theory, do $ \langle\mathbb{Z}, +\rangle $ and $ \langle\mathbb{R}, +\rangle $ have the same order?

Of course, $ \mathbb{Z} $ is countable, while $ \mathbb{R}$ is uncountable, so the two cannot be isomorphic. However, the reason for my question is the notation for the order of $ $$ ...
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0answers
29 views

Show that Z*n is the maximal subset of Zn which is a group with the operation [a] · [b] = [ab].

Let Zn be the set of equivalence class of integers mod(n) which are relatively prime to n, i.e. Zn = 􏰀[i] | gcd(n, i) = 1􏰁. So I understand how this works when the binary operation is ...
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1answer
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Show that $S_4$ is not isomorphic to $D_4\times\mathbb{Z}_3$

Show that $S_4$ is not isomorphic to $D_4\times\mathbb{Z}_3$ I have no idea how to show this. I'm studying for a test, so I am less interested in solutions and hints than I am strategy. What ...
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2answers
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Formally real field with two different orders

If $F$ is a formally real field, then there exists a total order relation on $F$ which is compatible with its sum and product, but it needs not be unique. a) How different can be two (compatible with ...
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1answer
23 views

Equivalent properties of Von Neumann regular rings

Let $M$ be a module over a ring $A$ and $R=Hom_A(M,M)$ its endomorphism ring (with respect to the composition). I need to show these following conditions are equivalent: $\alpha = \alpha \beta ...
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63 views

Show that $\phi(p^e)=p^e-p^{e-1}$

In an exercise I was asked to show that if $R$ is a ring with relatively prime ideals $I_1,I_2$ then $R/I \cong R/I_1 \oplus R/I_2$ where $I=I_1 \cap I_2$ and $\oplus$ is the direct sum. A follow on ...
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1answer
36 views

Name of and references for the equivalence relation $x \sim y :\Longleftrightarrow x^2 = y^2$

Playing around with the concepts of negativity and positivity, I came across the following equivalence relation defined for all elements $x,y$ of a field $\mathbb{F}$: $$ x \sim y :\Longleftrightarrow ...
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1answer
29 views

If I is an irreducible ideal, and P is a prime ideal, is (I+P)/P irreducible?

Let $A$ be a commutative ring with unit, and $P$ a prime ideal. My question is: If $I$ is an irreducible ideal in $A$, is $(I+P)/P$ irreducible in $A/P$? If not, can you show a counterexample? ...
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1answer
41 views

Prove that a group of infinite order must have a proper subgroup [on hold]

Assume that the group of infinite order is also cylic. How would one prove that? I am quite stuck.
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permutations and representations , sign function.

Show that the sign representation of $S_n$ is indeed a representation. attempt: Recall the sign function of a permutation is given by $\mathrm{sgn}(\pi) = (-1)^k$. Then recall a representation is a ...
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2answers
48 views

Roots of $x^p + x + [\alpha]_p \in \mathbb{Z}_p[x]$

Let $$g(x) = x^p + x + [\alpha]_p \in \mathbb{Z}_p[x],$$ where $p$ is prime. For which $\alpha, p \in \mathbb{Z}$ does $g(x)$ have at least one root? And for which $\alpha, p \in \mathbb{Z}$ ...
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1answer
37 views

How can i prove that a group of order 60 Is not simple? [on hold]

How can i prove that a group of order 60 Is not simple? please help me guys . I m fed up to think about this
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14answers
4k views

Can an infinite sum of irrational numbers be rational?

Let $S = \sum_ {k=1}^\infty a_k $ where each $a_k$ is positive and irrational. Is it possible for $S$ to be rational, considering the additional restriction that non of the $a_k$'s is a linear ...
0
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1answer
19 views

$A_n$ is generated by 3-cycles given $n\geq 3$. Is this proof correct?

The elements of $A_n$ is either of the form $(a,b,c,...)...$ or of the form $(a,b)(c,d)...$ In both cases, the element is a product of an even number of transpositions, not pairwise disjoint in the ...
0
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1answer
10 views

In $\mathbb{Z}_{79}$, $(\alpha\gamma + 66\beta\delta, \alpha\delta + 6\beta\gamma) = (0,0) \implies (\gamma, \delta) = (0,0)$ [on hold]

How does one prove that, in $\mathbb{Z}_{79}$, if $(\alpha, \beta) \neq (0,0)$, then $$(\alpha\gamma + 66\beta\delta, \alpha\delta + 6\beta\gamma) = (0,0) \implies (\gamma, \delta) = (0,0).$$ This ...
1
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1answer
26 views

Is every representation of $G$ over $\mathbb{K}$ trivial?

True/False: Let G be a group with $|G| = p^n$ for a prime number $p$ and $n \in \mathbb{N}$, and let $\mathbb{K}$ be a field of characteristic $p$. Is every representation of $G$ over $\mathbb{K}$ ...
2
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1answer
40 views

Normalizers of subgroups of Sylow $p$-subgroups

I am wondering whether there is an easy example of a finite group $G$ with a Sylow $p$-subgroup $P$ and a subgroup $Q\leq P$ such that the normalizer $N_P(Q)$ of $Q$ in $P$ is NOT a Sylow $p$-subgroup ...