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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trace of left/right multiplication

Let $A$ be a finite dimensional algebra over some field $k$. Then any element $a \in A$ defines two $k$-linear maps $a_l, a_r$ on $A$ by left and right multiplication, respectively. So there are two ...
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35 views

Proving a fundamental group is NOT abelian

I was wondering if the following approach would be possible in proving the fundamental group of $X$ was not abelian. If one can show there exists a homomorphism: $\pi_1(X, x_0) \rightarrow ...
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32 views

Conjugacy classes of $S_n$ under the action of $S_{n-1}$

I try to get explicitly сonjugacy classes of $S_n$ under the action of $S_{n-1}$. I believe that in the description of the classes present cycle type of a permutation and yet another parameter. But I ...
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33 views

Non-existence of non-abelian simple groups of order $90$ by counting elements [duplicate]

I want to show that no group of order $90=2 \cdot3^2 \cdot5 $ is simple. Part (iii) of Sylow's Theorem gives $$n_2 \in\{1,3,5,9,15,45\} $$ $$n_3 \in \{1,10\} $$ $$n_5 \in \{1,6\} $$ I am aware of ...
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Splitting Field Problem: what is $[S:\mathbb Q]$? [closed]

Find the splitting field $S$ of the polynomial $x^3-3x-1$ over $\mathbb Q$ and what is $[S:\mathbb Q]$?
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Calculate the Kernel of $D$ with char $F=p$

Let $F$ be a ring and $f(x) = a_0 + a_1x + · · · + a_nx^n$ be in $F[x]$. Define $f'(x) = a_1 + 2a_2x + · · · + na_nx^n−1$ to be the derivative of $f(x)$. we can define a homomorphism of abelian ...
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Suppose that $a$ and $b$ belong to a commutative ring $R$. If $a$ is a unit of $R$ and $b^{2}=0$ . Show that $a+b$ is a unit of $R$

Suppose that $a$ and $b$ belong to a commutative ring $R$. If $a$ is a unit of $R$ and $b^{2}=0$ . Show that $a+b$ is a unit of $R$ I try it.. consider $(a+b)(b)=ab+b^{2}=ab=1$ Since a is unit
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27 views

Let $F|K$ be a field extension and $a \in F $ such that $[K(a):K]$ is odd integer [duplicate]

Let $F|K$ be a field extension and $a \in F$ such that $[K(a):K]$ is odd integer, then prove that $K(a)=K(a^2)$.
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Find an element of order $45$ in the group $\mathbb{Z}_{30}\oplus\mathbb{Z}_{12}$, or explain why it is impossible

I'm asked to find the object asked for, or explain why it is impossible. Any help would be appreciated. Thanks for your time.
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Representatives of simple $\mathbb C[\mathbb D_3]$-modules (left modules)

Problem Let $\mathbb D_3$ be the symmetry group of the equilateral triangle. Give a complete list of the representatives of the simple left $\mathbb C[\mathbb D_3]$-modules. My attempt at a solution ...
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64 views

An example of free group

Let $\alpha : \mathbb{C}\cup\{\infty\} \to \mathbb{C}\cup\{\infty\}$ with $\alpha(x)=x+2$ and $\beta:\mathbb{C}\cup\{\infty\} \to \mathbb{C}\cup\{\infty\}$ with $ \beta(x)=x/(2x+1)$. Show that the ...
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1answer
145 views

Flexes of cubic curve

Which are the flexes of the cubic curve of Fermat $$x^3+y^3+z^3=0$$ at $\mathbb{P}^2(\mathbb{C})$ ? Could you give me a hint how we could find the flexes? Do we have to use maybe the following ...
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63 views

Are any of these rings isomorphic?

As part of my ongoing struggle to understand the complex conics, I've reached the following problem: Let $Q_1 = x^2 + y^2$, $Q_2 = x^2 - 1$, and $Q_3 = x^2$ be polynomials in $\mathbb{C}[x,y]$. ...
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44 views

Can a multiplicatively closed subset contain zero?

Let $A$ be a ring and $S$ be a multiplicatively closed subset. Can $S$ contain $0$? If so what will happen if we do $S^{-1}A$? A concrete and easy example coming to my mind is $A = \Bbb Z$, and $S = ...
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37 views

Algebra - Group Theory help… Intersection notation

How do you write $\{g ∈ G : \mu (Hx,g) = Hx,∀x∈G \}$ in intersection form? where $\mu (Hx,g) = Hgx$
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elements of order $3$ in $A_n$

Let $S_n$ be the permutation group on $n$. We know that $A_n \trianglelefteq S_n$. How many elements $ a \in A_5$ have order three. Is there any formula for finding number of elements in $ S_n $ or ...
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26 views

Odd dimensional universal quadratic form is isotropic?

For odd dimensional nondegenerate universal form, is it isotropic? All isotropic form is universal, but I wonder reverse case. I try to break it down into single form and even dimensional form but it ...
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0answers
27 views

For an algebra M over a ring A, why does the ring A need to be commutative? [duplicate]

All the definitions for an algebra over a ring I see begin with 'Let A be a commutative ring...' Why is this important? I've never seen it explained...
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7 views

Purely inseparable extension from Hungerford

Hungerford, Algebra, V.6.4 says, $F/K$ is purely inseparable if and only if $F$ is generated by a set of purely inseparable elements over $K$. My question : is there any purely inseparable extension ...
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26 views

Find number of all $a \in G $ such that $o(a) =3$

Let $G$ be a group and $|G|= 51$ find number of all $a \in G$ such that $o(a)=3$ My solution : by this theorem : if $|G|=pq$ that $ p ,q$ are prime. If $ q\nmid p-1 $ then $\quad$ $G \cong \Bbb ...
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Trying to understand group presentations using the example of the Dihedral group

According to Wikipedia the Dihedral group $D_n \cong \; \langle r,s \mid r^n = 1, s^2 = 1, s^{-1}rs = r^{-1}\rangle$. But why does this apply? As far as I understand the group presentation means that ...
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28 views

Order of abelian groups

Suppose $G = \mathbb{Z}_2 * \mathbb{Z}_4$. Then one can conclude that $G/[G, G]$ (the abelianization of $G$) is equivalent to $\mathbb{Z}_2 \oplus \mathbb{Z}_4$ because these are both abelian groups. ...
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No invariant complement?

How do I show that the representation $\rho: \mathbb{Z} \to \text{GL}_2(\mathbb{C})$ with $$\rho(1) = \begin{pmatrix}1 & 1\\ 0 & 1\end{pmatrix}$$ has an invariant subspace with no invariant ...
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42 views

No character theory, any representation ${\bf{GL}}_N(\mathbb{C})$ is reducible, upper bound

Let $G$ be any finite group. How do I show without character theory that there is a number $N = N(G)$ so that any representation $\rho: G \to {\bf{GL}}_N(\mathbb{C})$ is reducible (and finding an ...
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57 views

Proving irreducibility of polynomial over various fields

I am working on a problem set from an online course and am struggling to understand a key concept related to proving reducibility / split-ability etc. in a finite field $\mathbb{F}_{p^n}$, as well as ...
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16 views

Invariant factors of finite abelian group

Calculate the invariant factors of the group $G=\mathbb Z_{12} \oplus \mathbb Z_{21} \oplus \mathbb Z \oplus \mathbb Z \oplus \mathbb Z_{20} \oplus \mathbb Z_{9} \oplus \mathbb Z_7$. Applying the ...
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243 views

endomorphism of finite groups

Have $\mathcal{G}$ denote the set of finite groups with at least $2$ elements. How would I go about showing that if $G \in \mathcal{G}$, then $\left|\text{End}(G)\right| \le ...
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34 views

Primitive Permutation Group and Centralizers

Automorphism group of the Alternating Group - a proof In the above question, Derek Holt asserts that a primitive permutation groups has trivial centraliser in the symmetric group. Since I could not ...
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31 views

Order of Abelianization

If $G = G_1 * G_2$ where $G_1$ and $G_2$ are both cyclic of degree $n$ and $m$, and we know that $G/[G, G]$ (i.e., the abelianization of $G$) is isomorphic to $(G_1 / [G_1, G_1]) \oplus (G_2 / [G_2, ...
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Describing integral closure of quadratic number fields

I'm facing the following problem. Let $p$ be a prime and $ K=\mathbb{Q}(\sqrt{p}) $. I'm trying to find the integral closure of $ \mathbb{Z} $ in $ K $. I don't really know where to start. I've ...
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39 views

$G$ finite abelian group, $p$ prime that divides order of $G$

Problem Let $G$ be a finite abelian group and $p$ a positive prime that divides $|G|$. Show that the number of elements of order $p$ in $G$ is coprime with $p$. Let $|G|=p^nm$ with $n \geq 1$ and ...
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$P+Q:=\varphi_{O}^{-1}\left(\varphi_{O}(P)+\varphi_{O}(Q)\right)$

let X is affine space and $\overrightarrow{X}$ is vector space associted to X $$\begin{array}{ccccc} & \varphi_{O} : & X & \longrightarrow & \overrightarrow{X}\\ & & ...
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1answer
27 views

Is {0} a free module?

Is $\{0\}$ a free module (over any ring $R$) ? A free module is isomorphic to $R^n$, but is $n=0$ allowed? Alternatively, a free module is defined to have a set of linearly independent generating ...
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Relatively prime polynomials over integrally closed domain are(?) relatively prime over the fraction field

I know that the following holds: Lemma. Let $R$ be an integrally closed domain and $K$ its field of fractions. Let $f \in R[X] \setminus R$ monic. Suppose that there exist $g,h \in K[X] \setminus ...
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Does the projective Fermat curve have singular points?

I want to check if the projective Fermat curve, $$X^n+Y^n+Z^n=0, n \geq 1$$ has singular points. Could you give me a hint how we could do this? Do we have to check maybe if the curve is ...
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Abelian group of order $p^2q^2$ ($p$,$q$ distinct primes) determine number of elements of order $pq$ and $pq^2$

Problem For each abelian group of order $p^2q^2$ determine the number of elements of order $pq$ and the number of elements of order $pq^2$ in $G$. By the structure theorem we have that $$(1) ...
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1answer
18 views

Proof with exact sequence of modules

I'm trying to prove that if the sequence $$ M \xrightarrow{\varphi} W \rightarrow 0$$ is exact with $ W $ being a free module, then $ M \simeq \ker{\varphi} \oplus W $ What I got is that since $ W ...
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1answer
36 views

Showing an automorphism

I am trying to show that $Aut(Z_4+Z_2)$ is the dihedral group $D_4$ or the Quaternions $Q$. The Quaternions have 1 element of order 2, while $D_4$ has 5 elements of order 2. So if I show two ...
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21 views

Homology of Derivations of a dgca algebra

Let $(A,d)$ be a differential graded commutative and associative algebra. A derivation on $A$ is a linear endomorphism $L: A \to A$, that satsfies $L(ab)= L(a)b+ aL(b)$. More general a derivation of ...
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How can cancellation laws be applied to deduce equality?

Let $G$ be a group and let $g,h, k$ belong to $G$. Show that the following statement is true: $$ghg=gkg \implies h=k$$ Show that the following statement is not necessarily true: $$hgh=kgk \implies ...
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Decompose $a = a_1\cdots a_k$ and $a_1 + \dots +a_k = 0$

Problem. Prove that in the field $F, \text{char }F\neq2$ every element $a$ can be decomposed in the following way: $a = a_1\cdots a_k$ and $a_1 + \dots +a_k = 0$. Attempt 1. For the fields in which ...
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Proof of an existence of an algebraic closure of a given field

I am studying the field theory with Abstract Algebra by Dummit & Foote. A proof of an existence of an algebraic closure by constructing such extension using the Zorn's lemma is given in this ...
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Does there exist a surjective homomorphism between every pair of monoids? [closed]

Say we have two monoids $N,M$ and w.l.o.g. assume that $|N| \geq |M|$. Does there exist a surjective homomorphism $\varphi : N \to M$?
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Finding the Galois Correspondence of polynomial $t^4-2$

For the polynomial $t^4-2$ in $\Bbb Q[t]$, the splitting field is given by $\Bbb Q(\alpha, i)$ where $\alpha$ is $2^{1/4}$. I figured out that the Galois group of this polynomial is the dihedral ...
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652 views

Is the hyperbola isomorphic to the circle?

Is the ring $B=\mathbb{C}[x,y]/(xy-1)$ isomorphic with $C=\mathbb{C}[x,y]/(x^2+y^2-1)$? I think they shouldn't but all my tryings fail to prove the fact. Are they in fact isomorphic so I may try to ...
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Why two extension fields are isomorphic as vector spaces but not fields?

I understand that $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$ are isomorphic as vector spaces but not as fields. However, I do not understand why that is true. What is happening when they are ...
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78 views

non-examples for Krull-Schmidt-Azumaya

I am looking for some rings $A$ and finitely generated $A$-modules $M$, where the conclusion of the Krull-Schmidt-Azumaya Theorem does not hold for $M$, i.e. where $M$ can be written as direct sums of ...
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Semisimple ring and finitely generated modules exercise

Problem Let $R$ be a semisimple ring (i.e. $R$ is semisimple as a left $R$-module). Show that if $M,N$ and $P$ are finitely generated $R$-modules such that $M \oplus P \cong N \oplus P$, then ...
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Order of center of a p-group deduce abelian [duplicate]

Let $G$ be a group of order $p^n$, $p$ a prime. Suppose the center of G has order at least $p^{(n−1)}$. Prove that G is abelian. Attempt: use the class equation $|G|=|Z(G)|+ \sum_{i \in ...
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Why the ideals here are in this form?

In this article, in the proof of problem no. 6, p. 3, it listed all the possible ideals, because they contains at least one of $2$, $3$, $5$, from $120$. And at least one of $x+1$, $x^2-x+1$. But I ...