Tagged Questions

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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5
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“Wheel Theory”, Extended Reals, Limits, and “Nullity”: Can DNE limits be made to equal the element “$0/0$”?

"Wheels" are a little-known kind of algebraic structure: They modify the concept of a field or a ring in such a way that division by any element is possible, including division by zero, while also ...
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0answers
42 views

Prove that module has finitely many elements

Let $p$ be a prime number. Consider the subring $U:= \mathbb{Z}[1/p]$ of $\mathbb{Q}$ and define the $\mathbb{Z}$-module $M:=U/ \mathbb{Z}$ (1): Show that any $\mathbb{Z}$-submodule of $M$ that is ...
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0answers
10 views

Cumulants in diagrammatics (without physics or probability theory)

Formal cumulants ($\kappa_n$) and the associated moments ($\mu_n^{'}$) are related through log and exp transformations of exponential generating functions (e.g.f.): $$\exp \left [ \sum_{n=1}^{\infty ...
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0answers
9 views

Orthogonality Relations for Character Groups

I'm trying to understand a part of a proof of orthogonality relations for character groups and finite abelian groups, and I don't quite get this part from the below link: ...
1
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1answer
40 views

In the definition of a quotient group, does the subgroup have to be normal?

Let $G$ be a group and let $H \leq G$. Does $H$ need to be a normal subgroup to have the quotient group $G/H$? Progress I think yes. By definition, a subgroup $H$ is normal if and only if it is ...
0
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1answer
27 views

How does a quaternion differ from a position in terms of algebraic structure?

For two positions, I can subtract one from another to get a vector; I can take combination of them to get another position. My question is, can I treat quaternions in the same way? To be more ...
1
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1answer
21 views

Does this group theory question require an additional hypothesis?

The problem is to show that if $G$ is a finite group and for all nontrivial elements $a, b$ there exists an automorphism taking $a$ to $b$, then $G$ is a $C_p$ vector space, where $C_p$ is the group ...
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2answers
31 views

Cosets: Prove $\mathbf{H} \leq \mathbf{G}$

Let $\mathbf{G} = GL_n (\mathbb{R})$ and $\mathbf{H} = \{ H \in \mathbf{G}: \det(H) = \pm1 \}$. I'm trying to prove that $\mathbf{H} \leq \mathbf{G}$. And then: Given $A$, $B \in \mathbf{G}$, ...
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1answer
15 views

Invertible Ideal in Noetherian Integral Domain

I have been spending hours reading proof to a lemma but am stuck, I would love to get some pointers from you gurus to get it going, see below. (The original proof is in one big paragraph but I ...
4
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0answers
54 views

Are there irreducible ideals that are not primary in $K[X_1,\dots,X_n,\dots]$?

I can give examples of non-noetherian rings having irreducible ideals that are not primary. Among them are idealizations and valuation domains. But the first non-noetherian ring we are thinking about ...
12
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2answers
589 views

What's wrong with this proof that commutativity is implied by the other field axioms?

I seem to have found a proof that the commutativity of $+$ follows from the other field axioms. It is as follows: Let $(k,+,\cdot)$ be a structure satisfying all field axioms except commutativity of ...
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0answers
13 views

centre of an infinite p group may be trivial. [duplicate]

I have to prove this and can not seem to find any such example. I know for finite groups, centre is always non trivial, but what example is there for infinite. Thanks
0
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1answer
33 views

Inverse of a $G$-equivariant map

Let $G$ be a finite group and $X$ a set that has a transitive left action. I am trying to show that $A := \{f: X \rightarrow X: f(g.x) = g.f(x)\}$ is a group. The only thing I'm having trouble showing ...
0
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2answers
21 views

Finding a normal subgroup

Let $A$, $B$ be groups. Find an example for a normal subgroup $C$ of $A \times B$ that is not the direct product of $A \cap C$ and $B \cap C$. I have some troubles to understand how this can be ...
0
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1answer
14 views

Comparing indices of groups

Let $G$ be a group, $A$,$B$ be subgroups of $G$ and $A \subseteq B \subseteq G$ so that $[G : A]$ and $[ B : A ]$ are finite. Show that $[G : A] = [G : B] \cdot [G : A]$. If $G$, $A$ and $B$ ...
2
votes
3answers
118 views

Identity and Inverse Homomorphisms

For a group G and an abelian group H, Hom(G,H) is the set of all homomorphisms from G to H. My notes from class talk about the identity and the inverse homomorphism- I was wondering what these are? ...
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2answers
36 views

$NH=KN\;$ and $\;H\cap N=K \cap N$ imply $H=K$

Let be $G$ a group and $H$ and $K$ two subgroups such that $H\leq K \leq G$. Let be $N\trianglelefteq G$. How can I prove that the relationions $NH=KN\;$ and $\;H\cap N=K \cap N$ imply $H=K$? ...
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0answers
13 views

Showing that a F[t]-module is cyclic [duplicate]

Let F be a field and V be a finite dimensional vector space over F. And Let F[t] be the polynomial ring. Fix a linear map T : V -> V. Now V can be viewed as an F[t]-module. My question is, if the ...
0
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1answer
42 views

Why are these two quotients equal?

I'm not being able to check why are these two quotients equal. $\mathbb C[x]/(x^2-x^3)= \mathbb C[x]/(x^2)$ Can someone tell me why is it valid?
0
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1answer
14 views

Show that map from $\text{Hom$_A$($_AA,M$)} \to M$ is injective

Let $A$ be a unital ring and $M\in\text{A-Mod}$ Then $\text{Hom$_A$($_AA,M$)} \cong M$ By $\phi \mapsto \phi(1)$ I want to show that this map is injective, without using Yoneda Lemma. I have a proof ...
2
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2answers
49 views

How to find a minimal polynomial

I need to find minimal polynomial of $\alpha = \sqrt 2 + \sqrt [3] 3 $ over $\mathbb Q$ and prove that my result is minimal polynomial. How do I do that?
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1answer
37 views

What is the definition of “Augmentation Ideal Filtration”?

Let $A$ be an algebra. What is the definition of the Augmentation Ideal Filtration of $A$? Any answer with reference will be greatly appreciated.
0
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1answer
29 views

Studying for Abstract Algebra: Group

Help me understand more the example of the book so I may understand the whole thing. I am more on detailed-solution-kind of student to make myself get the idea. I am new to this topic and I find ...
1
vote
2answers
50 views

How to find the order of a group generated by two elements?

What is the order of a group $G $ generated by two elements $x$ and $y$ subject only to the relations $x^3 = y^2 = (xy)^2 = 1$? List the subgroups of $G$. Since the above relation is the 'only' ...
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1answer
34 views

Why lower central series imply $N$ series?

Let $G$ be a group and let $H_1,H_2,\cdots$ be subgroups of $G$ such that $$G=H_1\supset H_2\supset\cdots,$$ where $H_{k+1}$ is normal in $H_k$ and $[H_k,H_l]\subset H_{k+l}$ for any $k,l\geq 1$. Such ...
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0answers
6 views

Number of elements of composite order in a group

What can we say about the structure of the group which has at most 5 elements of composite order? For instance, can we present an upper bound for the order of such a group?
0
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1answer
23 views

Subgroups of isometries on Euclidean space

I am solving the following exercise: Let $\mathcal{T}(E) := \{ T_v \mid v \in \mathbb{R^2} \}$ be the set of all translations of E and $\mathcal{O}(2,\mathbb{R}) := \{g \in ...
2
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0answers
17 views

Reference Request: Johnson Filtration

I need to learn the Johnson Filtration, which I believe is defined on the automorphism group of free groups. Can anyone recommend some reference to this topic? I know one paper called "On the ...
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0answers
19 views

Ideals in a polynomial ring over a skew field

I know that a polynomial ring over a field is a PID, does this property also hold for a polynomial ring over a skew field? Is there maybe something else that characterise the ideals in that ring ?
0
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1answer
15 views

Induced homomorphism, first isomorphism theorem

Can someone explain to me what an induced homomorphism is? This is pertaining to the first isomorphism theorem in abstract algebra. If we have a function $f$ that maps group $G$ to ...
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1answer
15 views

Existence of Irreducible Character s.t. $\chi(g) \neq 0, \chi(1) \neq 0 \text{ mod } |C(g)|$ for Elements in Conjugacy Class of Prime Order

Given a finite group $G$, and a non-identity representative $g$ in a conjugacy class of prime order $p$, I'm trying to show that some nontrivial irreducible character of $G$ must have $\chi(g) \neq 0$ ...
1
vote
1answer
17 views

Confused about simple and semisimple modules

I was reading one of the possible definitions of semisimple modules, which is "for every submodule $N$ of $M$, there exists a submodule $P$ such that $M=N \bigoplus P$. After reading this I ...
2
votes
2answers
44 views

How to show that there is no odd $n$ bigger than 100 such that the group $U(n)$ will contain $2^n$ number of elements?

Consider $U(n):=\{1\leq r\leq n: (r, n)=1\}$. Under multiplication modulo $n$ it forms an abelian group. Its order will be $\varphi(n)$ i.e. $\varphi(n)=|U(n)|$. Assume that $n\geq 100$. Then my ...
0
votes
1answer
28 views

Cosets of $SL^\pm (n,\mathbb{R})$

Here is the question I am struggling with. If anyone could just give a push in the right direction it would be greatly appreciated! Let $\mathbf{H} = SL^\pm (n,\mathbb{R})$ Given $A$, $B \in ...
0
votes
1answer
16 views

Finding the kernel of a linear transformation

Let $x_1$ be an element of $W_1$, $x_2$ be an element of $W_2$, and $W_1$ & $W_2$ be subspaces of $V$. Consider the linear transformation $T$ given by $(x_1,x_2) \mapsto x_1 - x_2$ What is the ...
0
votes
1answer
21 views

Lower central series for $GL(2,\mathbb{C})$

how can I write the lower central series (http://en.wikipedia.org/wiki/Central_series#Lower_central_series) for $GL(2,\mathbb{C})$ ? I appreciate any help, because I do not clearly see where to ...
0
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0answers
11 views

Prove the subgroup <h.v> generated by h and v is Abelian and of order 9

I'm working on flushing out Theorems, Lemmas, Definitions, etc for a paper on the Rubik's Slide. (So I'll probably have more questions to come.) $\underline {What\ I\ Need\ To\ Know:}$ How to show ...
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0answers
8 views

Let G = A4 and suppose that G acts on itself by conjugation; that is, (g,h) → ghg^-1

(a) Determine the conjugacy classes (orbits) of each element of G. (b) Determine all of the isotropy subgroups for each element of G. I was fine with conjunction classes until it said "on itself" ...
4
votes
1answer
35 views

Question about what Z/6Z actually means?

I have an abstract algebra exam tomorrow, and I'm having a little bit of difficulty deciphering the difference between Z/6Z, Z6 and 6Z. Can someone please explain this to me? As far as I thought, Z6 ...
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2answers
14 views

Questions on: $g(f(x))=m(f(x))\implies g=m$ & $a(b(x))=a(c(x))\implies b=c$

Question: Are either of the below statements true? First equivalency: $g(f(x))=m(f(x))\implies g=m$ Second equivalency: $a(b(x))=a(c(x))\implies b=c$ I believe that the first is true, but ...
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1answer
43 views

A question about the quotient group $GL(2,\mathbb{C})/\{\lambda I\}$

My question is that : Is $GL(2,\mathbb{C})/\{\lambda I\}$ isomorphic to $PSL(2,\mathbb{C})$ ? If yes, how can one prove it ? I was thinking in this direction : Given any arbitrary matrix $A$ in ...
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1answer
40 views

Questions Regarding a Ring

I am extremely new to abstract math. I was given the following problem and below each of the questions, I have my answer. I can't imagine it is right because I am so confused. Please point me in the ...
0
votes
1answer
34 views

Ring/Nilpotent Proof [duplicate]

Let $R$ be a ring with unity, and suppose $x\in R$ is nilpotent $(i.e. x^n=0$ for some positive integer $n$ $)$. Prove that $1-x$ is a unit in $R$. Any hints or proofs are greatly appreciated. Rings ...
0
votes
1answer
23 views

$G$ is a group. $H\leq G$ and $K\trianglelefteq G$. Prove $HK\leq G$.

I tried to use the following but $H$ would have needed to be a normal subgroup of $G$. $$h_1k_1,h_2k_2\in HK\Rightarrow (h_1k_1)(h_2k_2)^{-1}\in HK.$$ How do you think I should prove it?
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1answer
29 views

Prove or disprove : $a_0+a_1 x+…+a_n x^n\in R[x]$ is nilpotent iff $a_i$ is nilpotent $\forall i$.

1) Prove or disprove : $a_0+a_1 x+...+a_n x^n\in R[x]$ is nilpotent iff $a_i$ is nilpotent $\forall i$. 2) Prove or disprove : $a_0+a_1 x+...+a_n x^n\in R[x]$ is a unit iff $a_i$ is nilpotent ...
3
votes
1answer
49 views

Morphisms of quasi-affine varieties and locally closed sub-varieties

The Problem Let $\varphi:X\to Y$ be a morphism of quasi-affine varieties. Let $Z\subset X$ be a locally closed sub-variety (that is, $Z$ is an open sub variety of a closed subvariety). Show that ...
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votes
2answers
33 views

Every subring of integers is of the form $m\mathbb Z$

Let $S$ be a subring of $\mathbb Z$. Prove that $S=m\mathbb Z$ for some $m\in \mathbb N_0$. I really have no clue. But if $S$ is a subring that means $S$ is a subset of $\mathbb Z$ and it ...
0
votes
2answers
55 views

Is it a wrong exercise in Pinter's Algebra?

It is the exercise D6 from chapter 8 of Pinter's Algebra. (I'm supposed to prove it.) Let $a$ and $b$ be cycles (not necessarily disjoint). If $a^2 = b^2$, then $a = b$. Am I correct that it is ...
3
votes
1answer
22 views

If $H,K$ are subgroups of $G$, and $G$ is finite, prove that $[K\colon (H\cap K)]\leq [G\colon H]$

Let $H,K$ be subgroups of a finite group $G$. Prove that $[K\colon (H\cap K)]\leq [G\colon H]$. This is what I have: $[K\colon (H\cap K)] = |\left\{ a(H\cap K) \mid a\in K\right\}|$ $[G\colon H] = ...
1
vote
1answer
27 views

Prove that polynomial ring $R[X][Y]$ over $R[X]$ is the same as ring of polynomials from 2 variables $R[X][Y]$.

I want to show that polynomial ring $R[X][Y]$ over $R[X]$ is the same as ring of polynomials from 2 variables $R[X][Y]$. Unfortunately I do not have intuition about first ring, so I do not see how to ...