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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Some subset is not a block in group action iff a separation property holds, questions on proof and special cases

(Separation Property) Suppose that $G$ is a group acting transitively on a set $\Omega$ with at least two points, and that $\Delta$ is a nonempty subset on $\Omega$. Show that $\Delta$ is not a ...
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18 views

$K, N < G$ and $|G:N|$ and $|K|$ coprime, then $K<N$. Group action argument?

A common exercise in finite group theory is Suppose $G$ is a finite group with $K < G$, $N \lhd G$. Suppose that $|K|$ and $|G : N|$ are coprime. Then $K < N$. The normal way to attack ...
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37 views

About $R/I$ Where $I$ is a Prime Ideal

A well known result in Commutative Algebra says: for a commutative ring $R$ with $1$, $R/I$ is an Integral Domain if and only if $I$ is a Prime Ideal of $R$. Can this result be generalised for non ...
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2answers
27 views

General way to find actions of automorphisms of the group of $x^3-2$ over $\mathbb{Q}$

I'm looking at the Galois group $\mathrm{G}(\mathbf{K}/\mathbb{Q})$ where $\mathbf{K}$ is the splitting field of $x^3-2$ over $\mathbb{Q}$. Of course, $\mathbf{K} = \mathbb{Q}(\alpha_1, i\sqrt{3})$, ...
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What is the meaning of “B is a bialgebra covariantly acting on A”?

Let $A$ be an algebra and $B$ a bialgebra. What is the meaning of "covariantly" in "B is covariantly acting on A"? Thank you very much. Edit: it is on line 13 of the abstract of the file (page 3).
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An iff involving finite dimensionality and maximal ideals

Let $R = \mathbb{C}[x_1, ... , x_n]$ and let $I$ be a non-zero ideal of $R$. Show that $R/I$ is a finite dimensional $\mathbb{C}$ algebra iff $I$ is contained in only finitely many maximal ideals of ...
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29 views

Closure of Set of Fractions with Lowest Terms Condition

Suppose I have a set of rational numbers where elements have denominators are odd and numerators and denominators are co-prime. I need to show that the set is closed under addition. It is clear that ...
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56 views

Why is Abstract Algebra so abstract? [on hold]

I mean, I haven't seen any concrete example about it. May you please give me one? Especially about Groups which we are currently having difficulties on. Thank you.
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3answers
73 views

Show that $\mathbb{Q}(\sqrt{2})$ is the smallest subfield of $\mathbb{C}$ that contains $\sqrt{2}$

This was an assertion made in our textbook but I have no idea how to show that either statement is true. Also would like to show that that $\mathbb{Q}(\sqrt{2})$ is strictly larger than $\mathbb{Q}$, ...
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Showing an iff about finitely generated modules over PID's

Let $R$ be a PID and $M$ be a finitely generated torsion $R$ module. Show that $M$ is a cyclic $R$ module iff for any prime $p \in R$ either $pM = M$ or $M/pM$ is a cyclic R module. Thoughts so far: ...
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1answer
48 views

$[G:H] < \infty$ then $gHg^{-1} = H$ and is it true that $gHg^{-1} = H$

G is a group. $H < G$ and $ g \in G$ $gHg^{-1} \subset H$ I need to prove the following : a) if $[G:H] < \infty$ then $gHg^{-1} = H$ b) without the additional fact given in (a) is it true ...
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1answer
53 views

$\mathbb{Q}\left(\sqrt{p}\right)=\{a+b\sqrt{p}\mid a, b \in\mathbb{Q}\}$ is a subfield of the field $\mathbb{R}$

Prove that $\mathbb{Q}\left(\sqrt{p}\right)=\{a+b\sqrt{p}\mid a, b \in\mathbb{Q}\}$ is a subfield of the field $\mathbb{R}$, where $p$ is a prime number I know this is true for many primes that ...
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3answers
40 views

Nilpotent ideal and ring homomorphism

In "Problems and Solutions in Mathematics", 2nd Edition, exercice 1308 Problem statement Let $I$ be a nilpotent ideal in a ring $R$, let $M$ and $N$ be $R$-modules, and let \begin{equation} f : M ...
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26 views

Finitely generated modules over principal ideal domain

Let A be principal ideal domain with field of fractions K. L is finite separable extension of K and B is integral closure of A in L. It is obvious that there exists constant d in A, such that dB is ...
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1answer
27 views

Find isomorphism between $D_6$ symmetries of regular triangle, $S_3$, and $GL_2(F_2)$

problem: Find isomorphism between $D_6$ symmetries of regular triangle, $S_3$, and $GL_2(F_2)$. I already proved $D_6$ is isomorphic to $S_3$. And $S_3 $ is isomorphic to $GL_2(F_2)$ Am I suppose ...
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36 views

Grothendiek and singular points

My question relates to the following thread I opened some weeks ago: A question regarding Grothendieck , topos and (adelic??) points Specifically, consider this paragraph: At 1:14:30 and after, ...
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$Y$ be the group such that it is define as $Y = < u, v | u^4 = v^3 = u= v=1, uv = v^2u^2, v^2 = v^{-1}>$ .

Let $Y$ be the group such that it is define as $Y = < u, v | u^4 = v^3 = u= v=1, uv = v^2u^2, v^2 = v^{-1}>$ . a) Show that $v$ commutes with $u^3$. [Show that $v^2u^3v = u^3$ by writing the ...
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Supersolubility of $ G/H $ and $ H $ not deduced supersolubility of $ G $.

I want show $ S_{4} $ isn't supersoluble group. For this suppose $ 1 \leq B_{4} \leq A_{4} \leq S_{4} $ be a normal serie of $ S_{4} $, that $ B_{4} $ is Klein’s four-group. Since $ B_{4} $ isn't ...
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30 views

rings of polynomials over $Z_p$ (part-2)

An element of R is a polynomial in $x$ of degree $< r$ with coefficients from $Z_n$ (where $n$ is a composite number). We use the notation $a(x)$ to represent elements of $R$. Let $\phi :R \mapsto ...
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2answers
28 views

What is the center of $\mathbb{C}S_3$?

How do I found the center of symmetric group algebra $\mathbb{C}S_3$? and in general $\mathbb{C}S_n$? I did an example on a smaller group algebra: $\mathbb{C}S_2=\{a (1)+b(12) \mid a,b\in ...
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33 views

Term rewriting; Compute critical pairs

I have tried to solve the following exercise but I got stuck while trying to find all the critical pairs. Any help is appreciated.
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1answer
16 views

Prove that the order of an element in $S_n$ equals the least common multiple of the lengths of the cycles in its cycle decomposition.

Prove that the order of an element in $S_n$ equals the least common multiple of the lengths of the cycles in its cycle decomposition. proof: Let $\sigma \in S_n$. Then $\sigma = ...
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1answer
43 views

A question about injective modules [on hold]

I need to find an injective module $B$ and a submodule $A$ of $B$ such that $B/A$ not injective.
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15 views

Is this true for quotients of Finitely Generated abelian groups?

Let $G\cong \mathbb{Z}_{{p_1}^{k_1}}\oplus\mathbb{Z}_{{p_2}^{k_2}}\oplus\dots\oplus\mathbb{Z}_{{p_n}^{k_n}}$, where the $p_i$'s are primes. Let $H\cong ...
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1answer
22 views

Stabilization of all even/odd terms of sequence of iterated centralizers.

This is related to my previous question, see here. Fix a ring $B$. Given a subring $A \subset B$, we define$$A^! := \{b \in B : ab = ba,\text{ }\forall\,a \in A\},$$the centralizer of $A$ in $B$. ...
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38 views

Question about sub-groups. [duplicate]

Let H,K be sub-groups of G (finite order), proof that if (G;H) and (G:K) are relatively Prime G=HK. Any clue?
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1answer
31 views

Chosen maximal subject is a subgroup

Let $ G $ is a finite soluble group and $ N $ be a unique minimal normal subgroup of $ G $. Let $ G = TS $ that $ S $ is the fitting subgroup of $ G $ and $ T = N_{G}(H) $ for $ H \leq G $. Suppose $ ...
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1answer
32 views

If two elements commute, does each element commutes with the inverse of the other [on hold]

Let $G$ be a group and $u,v \in G$. Is it possible that $uv = vu$ but $u^{-1}v \ne vu^{-1}$?
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2answers
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Sylow p-subgroup of order p does not normalize any other Sylow p-subgroup

Let $P_1,P_2$ be distinct Sylow p-subgroups of $G$ with order $p$. Is it generally true that $P_1$ cannot normalize $P_2$? I've seen algebra textbooks use this fact for $p=3,11$ and they quote 'a ...
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1answer
34 views

Subgroups of generalized dihedral groups

A generalized dihedral group, $D(H) := H \rtimes C_2$, is the semi-direct product of an abelian group $H$ with a cyclic group of order $2$, where $C_2$ acts on $H$ by inverting elements. I know that ...
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1answer
32 views

Product and intersection of ideals in a polynomial ring

I want to show that in the polynomial ring $K[X,Y,Z,W]$ (where $K$ is a field) the equality $(X,Y)\cap(Z,W)=(XZ,XW,YZ,YW)$ holds. Obviously RHS is contained in LHS. How to show that LHS is ...
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1answer
19 views

rings of polynomials over $Z_p$

An element of R is a polynomial in $x$ of degree $< r$ with coefficients from $Z_p$ (where $p$ is a prime). We use the notation $a(x)$ to represent elements of $R$. Define map $\phi :R \mapsto R$ ...
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Extension of group

Let $ p = 2 $, and Let $ H $ be a subgroup of a direct product of copies of $ S_{3} $. Why $ H $ is an extension of a $ 3 $-group by a $ 2 $-group, and $ H/O_{p^{\prime}}(H) $ is a $ 2 $-group ?
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A more formal but intuitive understanding (on a definition) of a group action

We know that a symmetric group $S_n$ acts on the set $\{1, 2,\ldots, n\}$. The definition of an action of a group $G$ on a set $S$ is a function $G\times S\to S$ such that: 1) $e\ast s=s$ 2) ...
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1answer
32 views

Find isomorphism between $S_3$ and $GL_2(F_2)$. [duplicate]

Find isomorphism between $S_3$ and $GL_2(F_2)$. proof: Let $A = \begin{pmatrix} a& b\\ c & d \end{pmatrix}$. Where $\det (A) \neq 0$. And recall $S_3$ is the permutation group with ...
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2answers
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Is $\mathbb{Q}(\alpha_i^2)$ an intermediate field of $\mathbf{K}/\mathbb{Q}$, where $K$ is the splitting field of $x^3-2$ over $\mathbb{Q}$?

I've found the Galois group of $x^3-2$, isomorphic to $\mathbf{S}_3$. It has 6 subgroups (including the trivial subgroup and the group itself), and thus by the Galois Correspondence there should be 6 ...
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1answer
23 views

Question in line of proof for first isomorphism theorem

Let $\phi: G_1 \to G_2$ be a group homomorphism. Let $\ker \left({\phi}\right)$ be the kernel of $\phi$. Then: $\operatorname {Im} \left({\phi}\right) \cong G_1 / \ker ...
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In Z_437, calculate 30 circled division 29 [on hold]

I am having trouble with modular division, especially finding inverses. I know the answer is 212, but I was hoping someone could show me how to reach this answer. Other practice problems include (all ...
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1answer
38 views

Field homomorphism induces an isomorphism between their prime subfields

So the question is: Let $\sigma$: $F_1 \xrightarrow[]{} F_2$ be a homomorphism where $F_1$ and $F_2$ are fields. Show $\sigma$ induces an isomorphism between their prime subfields and, in ...
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What's the importance of proving that $0,1$ are unique?

I had a course in the construction of numbers last semester. I understand the potencial of most of the proofs, for example: I guess I can answer decently why commutativity is important. But when it ...
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2answers
42 views

Let $f:Z \times Z \to Z$ with $f(1,1)=2$ and $f(3,5)=6$. Estimate the $\ker f$ of $f$ and $f(0,5)$

Let $f:Z \times Z \to Z$ with $f(1,1)=2$ and $f(3,5)=6$. Estimate the $\ker f$ of $f$ and $f(0,5)$. I am trying to solve this but i need any ideas or hints to start,any help would be interesting.
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Linear Algebra help needed [on hold]

It's been a while since I've taken linear algebra and I am trying to figure out the problem below. I am not sure how to start. Thank you for any and all help. I am not sure how to For any real ...
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29 views

Determining the kernel of a module homomorphism

Let $p$ be a prime and let $n$ be a positive integer such that $p^n > 2$. Set $R:= \mathbb{Z}_{p^n}$, that is, the residue ring with binary operations of addition and multiplication modulo $p^n$. ...
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1answer
25 views

Show $R \setminus S$ is a union of prime ideals

I'm stuck on the following question: Let $R$ be a commutative ring with $1$, and $S \subseteq R$ a saturated multiplicative set (that is, $1 \in S$ and $x, y \in S$ if and only if $xy \in S$). ...
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25 views

Hanging a painting with nails so that removing any subset of nails from a given collection makes painting fall, and subsets are minimal

So I'm aware of the result that for positive integers $k \leq n$ it's possible to hang a painting with $n$ nails, such that if any $k$ nails are removed then the painting falls, but never when $k-1$ ...
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1answer
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$ G/N $ is is a subgroup of a direct product of copies of the cyclic group $ C_{p-1} $ if $ p>2 $

Let $ G $ is soluble group and $ A $ be a unique minimal normal subgroup of $ G $. Then $ A $ is a elementary abelian group of a prime power. Let every chief factor of $ G/A $ has order $ 4 $ or a ...
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22 views

How to show #Hom$(C_a, G)=\{x\in G: x^a=e\}$?

I am willing to establish that $$\#\text{Hom}(C_a, G)=\#\{x\in G: x^a=e\}$$ where $G$ is finite group of order $n$ and $C_a$ is cyclic group of order $a$. I started like this: By first isomorphism ...
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4answers
60 views

$\mathbb{Z}_6/\mathbb{Z}_2$ isomorphic to $\mathbb{Z}_3$?

Recently in class my teacher mentioned that the quotient group $\mathbb{Z}_6/\mathbb{Z}_2$ is isomorphic to $\mathbb{Z}_3$. May I ask why is this so? Also, what do elements in ...
4
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1answer
27 views

Is a subring contained in the centralizer of its centralizer?

Fix a ring $B$. Given a subring $A \subset B$, we define$$A^! := \{b \in B : ab = ba,\text{ }\forall\,a \in A\},$$the centralizer of $A$ in $B$. This is a subring of $A$, so we can iterate $A^{!!} := ...
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1answer
22 views

If $G_1/N \unlhd G/N$ then $G_1 \unlhd G$?

I want to show that if $N$ is a simple normal subgroup of a group $G$ such that $G/N$ has a composition series, then also $G$ has a composition series. I think I can finish the proof if I can only ...