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

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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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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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 ...
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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
12 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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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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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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20 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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60 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
26 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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37 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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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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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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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 ...
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1answer
18 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 ...
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8 views

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

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 ...
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1answer
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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 $\mathbb{K}$ ...
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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 ...
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1answer
42 views

Quotient of direct sum of abelian groups

Let $A \oplus B \simeq A' \oplus B $. Does it follow that $A\simeq A'$? Many thanks in advance!
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references of discrete association scheme

I tried to find a book or paper to understanding discrete association scheme but I could not get any book for that. What is the good references for that?
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Show $p(x)$ is a primitive polynomial

First the definition: Polynomial $q(x) \in \mathbb{Z}_p[x]$ of degree $n$ is called primitive, iff: $q(x) \mid x^{p^n-1}-1$ $\forall k : 1 \leq k \leq p^{n}-1$ : $q(x) \nmid x^k - 1$ ...
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Noncommutative rings and the evaluation homomorphism

Recall the evaluation homomorphism of a ring. For example, if $\{R[x]|{p(x)=a_0+a_1x+a_2x^2...}$} is the ring of polynomials with real coefficients then we can evaluate with respect to $c$ by letting ...
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1answer
50 views

group action same thing as homomorphism

A linear group action of a group $G$ on a vector space $V$ is the same thing as a homomorphism from G to the general linear group $GL(V)$. attempt: Suppose a linear group action of a group $G$ on a ...
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1answer
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Order of element in polynomial ring in Hatcher

So I've been reading Hatcher and I am unsure what they mean when they say things like $H^\ast(\mathbb{R}P^n;\mathbb{Z}_2)\cong \mathbb{Z}_2[\alpha]/(\alpha^{n+1})$ where $|\alpha|=1$. It is this last ...
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1answer
74 views

If $f:\mathbb{Z} \to \mathbb{Z}$ is an isomorphism, prove that $f$ is the identity map. [on hold]

I am a little baffled by this question. Is it safe to assume that since $f$ is an isomorphism, $f (1) = 1$ ? And, if it is safe to assume this, could I construct a proof by induction, by using the ...
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2answers
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Let $\phi: K(X) \rightarrow K(X)$ be the K-morphism such that $\phi (X)=1-X$. Find an element $Y \in K(X)$ so that $K(Y)=\{ f\in K(X) : \phi (f)=f\}$.

Let $K(X)$ be the field of rational functions of $X$ over some field $K$. Let $\phi: K(X) \rightarrow K(X)$ be the K-morphism such that $\phi (X)=1-X$. We have $L:=\{ f\in K(X) : \phi (f)=f\}$. Find ...
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Verifying formulas for reflection, rotation, and translation in the Complex plane.

1) If $T_a : \Bbb C → \Bbb C$ is given by $T_a(z) = z + a$ then $T^{−1} _a (z) = z − a = T_{−a}(z)$ for some fixed $a ∈ \Bbb C$. 2) If $ρ_θ : \Bbb C → \Bbb C$ is given by $ρ_θ(z) = e^{iθ}z + a$ then ...
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HW - Number of subspaces T of a vector space K containing a fixed subspace M.

Given a vector space $K$ of dimension $k$ over a finite field $\mathbb{F}_q$, what is the number of subspaces $T$ of dimension $t<k$ that contain a given subspace $M$ of dimension $m<t$? ...
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For which $n \in \mathbb{N}$ does $x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n = [0]_n$ have at least $7$ distinct solutions?

I have to find one $n \in \mathbb{N}$ such that $$x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n = [0]_n$$ has at least $7$ distinct solutions in $\mathbb{Z}_n$ (or, equivalently, $f(x) = x^8 + ...
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If $G$ acts $k$-transitive and $k > 5$ and $G$ is neither alternating nor symmetric, then $(n-k)! \ge 2n$

The following is an exercise from D. Robinson: A Course in the Theory of Groups. Let $G$ be a $k$-transitive permutation group of degree $n$ which is neither alternating nor symmetric. Assume $k ...
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Centralizer of $(1,2,3)(4,5,6,7,8) \in A_{11}$

Let $$\tau = (1,2,3)(4,5,6,7,8) \in A_{11},$$ where $A_{11}$ is the group of even permutations. Let $H$ be the centralizer of $\tau$. I can easily prove that $H$ is a subgroup of $A_{11}$. How do I ...
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1answer
33 views

For which $p$ is $G_p = \{\sigma \in S_{12}: \text{ord}(\sigma) \text{ divides } p \}$ a subgroup of $S_{12}$?

Let $p$ be a prime number. I've to figure out for which $p$ $$G_p = \{\sigma \in S_{12}: \text{ord}(\sigma) \text{ divides } p \}$$ is a subgroup of $S_{12}$. I realize that we have to ...
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1answer
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Group isomorphism between two groups .

Let $SO_2(\mathbb{R})$ be the group of rotations of the circle under the operation of composition. And consider $\mathbb{R}$ as an additive group. Prove that $SO_2(\mathbb{R}) \cong \begin{Bmatrix} ...
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1answer
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Local properties of morphisms of schemes

In Hartshorne Proposition II.5.8, he shows, given a morphism $f \colon X \to Y$ where X and Y are schemes and $\mathcal{G}$ a quasi-coherent sheaf of $\mathcal{O}_{Y}$- modules, that ...
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1answer
35 views

Alternate proof to number of monomials in a given degree - “more” rigorous, formal [duplicate]

Let $s$ be the number of variables and $n$ be the degree of the monomials we want to count in $R[X_1,\dots,X_s]$. Then show, that the count is $$\delta(n,s):=\binom{s-1+n}{s-1}.$$ The question ...
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3answers
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Understanding the definition of the sign of a permutation , $\operatorname{sgn}(\pi) = (-1)^k$ .

I am trying to understand the definition of the sign of a permutation $\pi$. My textbook only mentions that $\operatorname{sgn}(\pi) = (-1)^k$ , where $k$ is the number of transpositions . But I ...
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1answer
20 views

Studying the symmetry group of the dodecahedron by introducing axes.

For my bachelor project I'm studying the symmetry of the Platonic solids (as a start at least). I computed the symmetry group of the tetrahedron by labeling the vertices, and the cube by labeling the ...
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2answers
17 views

Degree of field extension using minimum polynomial

Let $K \subset L$ an algebraic extension. I want to prove for $a,b \in L$ that $$[K(b): K] \geq [K(a,b): K(a)],$$ Where for example $[K(b):K]$ is the degree of the field extension $K \subset K(b)$. ...
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Proving an example on Sum Of Ideals [on hold]

Show that $A$ is an ideal of $A+B$ Show that $A+A=A$ for some ideal $A$ in a ring $R$
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40 views

subgroup having index $2$ of $R^*$

The question is to find all the subgroups of $R^*$ (non-zero reals under multiplication) of index $2$. The index can be found out for finite groups. How to find subgroups having certain index for an ...
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Finding subgroups with a specific property.

I'm trying to find subgroups with the following properties: if $[G:H]=n$ there is a $g∈G$ so that $g^n≠e$. What do I do know is that $H$ cannot be normal (previous exercise). I just can't find any ...
3
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1answer
61 views

$\Bbb R,\Bbb C,\Bbb H$ are the only complete normed division rings

Does anyone have a proof or reference for this often-quoted fact? How complicated is the proof? If $K$ is a complete normed division ring, then $K$ is isomorphic to the real numbers, complex ...