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Questions tagged [abstract-algebra]

Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, and modules, among other topics. Whenever you use this tag, please also include related tags like group-theory, ring-theory, modules, etc., in order to clarify ...

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Show that the factor ring $R[x]/(1−ax)$ is a zero ring, that is, $R[x]/(1−ax)=\{(1−ax)\}$

Let $R$ be a commutative ring with identity and let $a\in R$ such that $a^{(n−1)}$ is not zero, but $a^n$ is zero for some positive integer $n$. Show that the factor ring $R[x]/(1−ax)$ is a zero ring, ...
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Show $ Gal(L|Q)\cong (Z/7Z)^*$ and determine $L_H $

Let $ζ_7 = e^{2πi/7}$ and consider the extension $L = Q(ζ_7)$ of $Q$. I want to prove $ Gal(L|Q)\cong (Z/7Z)^*$ and also given $H$ is a subgroup of $G$ with $|H| = 2$, I want to determine $L_H$ . The ...
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About roots of unity in an arbitrary CM field K with an abelian extension L

I have an arbitrary CM field $K$ and an abelian extension $L$ of $K$ with dimension $[L:K]=p$, with $p$ prim. Suppose there exist a non trivial primitiv root of unity $\xi$ in K. Must $\xi$ be a $p$-...
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17 views

Connected Lie group 1-dimensional isomorphic to $\mathbb{R}$ or to $S^1$

Let $G$ a connected Lie group of dimension 1. Show that \begin{align} G \cong \mathbb{R} \, \, \, \text{or} \, \, \, G \cong S^1 \end{align} I tried to read and understand the topic Connected, one-...
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8 views

Composition factors above a subnormal subgroup

Suppose $G$ has a composition series and let $H$ be a subnormal subgroup of $G$. Then $H$ is a term of a composition series of $G$. I know that the number of terms above $H$ in any composition series ...
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olympiad level question

let p(x) = x^2+ax+b such that p(x)=0 and p(p(p(x)))=0 has equal roots find the integral value of p(0).p(1)? i tried it a lot but not done.. PLEASE HELP ME
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What is the name for an algebraic structure with closure and identity? [duplicate]

What is the name for the algebraic structure that has closure and an identity element? You could call it a group without associativity or inverses. Does this structure have a name?
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22 views

Deduce that the symmetry group of the dodecahedron is a subgroup of $S_5$ of order 60.

Let D be a regular dodecahedron. It is possible to inscribe a cube on the vertices of D thus: (a) Prove that one can inscribe exactly 5 such cubes inside D. (b) Deduce that any rigid motion of D (i....
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1answer
34 views

Prove that $(-1)^{a/b}= \cos(180(b-a)/b)+i \sin(180(b-a)/b)$

I wonder why $(-1)^{a/b}= \cos(180(b-a)/b)+i \sin(180(b-a)/b)$ I got a feeling that this has something to do with complex plane.
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how can I prove that $p=7,n=2$ is the only solution (sum of divisors)?

Question: Find every pair of $(n,p)$ in which $n$ is a positive integer and $p$ is an odd prime number so that the sum of every positive divisor of $p^{2^n-1}$ is a square number. It can be seen that ...
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2answers
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Elements of $S_n$ can be written as a product of $k$-cycles.

Let $k\leq n$ be even. Prove that every element in $S_n$ can be written as a product of $k$-cycles. I really have no idea how to go about this. My initial intuition was to proceed by induction first ...
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2answers
39 views

$(1-ba)$ left-invertible $\implies (1-ab)$ left-invertible

Suppose $(1-ba)$ in a ring $R$ is left-invertible. Then we wish to show that $(1-ba)$ is left invertible and explicitly construct its inverse. We have $Rb(1-ab)=R(1-ba)b=Rb \subseteq R(1-ab)$. I'm ...
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1answer
23 views

Some results given that $|G| = nm$

Let $G$ be a finite group such that $|G| = nm$. Suppose $x\in G$ has order $n$ and let $\sigma_x\in S_G$ be the permutation such that $\sigma_x(g)=xg$ for every $g\in G$. Note that $\sigma_x$ is a ...
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1answer
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Understanding conjugacy classes in $SL_{2}(\mathbb{F}_{q})$

I am trying to calculate the conjugacy classes of the group $SL_{2}(\mathbb{F}_{q})$, with the help of the knowledge of conjugacy classes of $GL_{2}(\mathbb{F}_{q})$. I am using two of the following ...
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If $A\cong B$, $C\cong D$ with $D\trianglelefteq B$, $C \trianglelefteq A$ then $A/C\cong B/D$.

Suppose that $A,B,C,D$ are groups such that $A\cong B$, $C\cong D$ with $D\trianglelefteq B$, $C \trianglelefteq A$. Prove that $A/C\cong B/D$. Proof: Suppose that $f:A\to B$ and $g:C\to D$ are ...
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1answer
28 views

Solution to a special type of polynomial equation

Let $k$ be any field (maybe need to be algebraically closed), and $n$ be a positive integer, $k[x_1,\ldots,x_n]$ be the polynomial ring. Then consider the equation $$\sum_{i=1}^n l_i(x)x_i^2=q(x)(x_1+\...
3
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1answer
45 views

Abelian group operation on $(0, 1)$

My Question is: is there a binary operation on $(0, 1)$ that makes this set into an abelian group in which the inverse for any $x$ is $1-x$? My approach to this was applying $f(x)=\pi(x-0.5)$ (which ...
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1answer
36 views

Prove that if $|a| = m$ and $|b| = n$, then $\exists g \in G$ s.t $|g| = mn$

In the book of Algebra by Hungerford, at page 36, question 2, it is asked that For the case where $(m,n) = 1$, I have done the following: Observe that $$(ab)^{mn} = a^{mn} b^{mn} = e.$$ Now, let $...
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0answers
21 views

Nullhomotopy of a chain map

Let $(C, \partial), (C',\partial')$ be two chain complexes, and $f: C \rightarrow C'$ be a chain map. In general we know that if $f$ is zero-homotopic (i.e. $\exists$ a collection of maps $s_k: C_k \...
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10 views

$F_{p^n}=\mathbb{Z}_p(\alpha)$, determine whether $\alpha$ is the generator of multiplicative group $F_{p^n}^*=F_p\backslash \{0\}$.

$F_{p^n}=\mathbb{Z}_p(\alpha)$, determine whether $\alpha$ is the generator of multiplicative group $F_{p^n}^*=F_p\backslash \{0\}$. My try: Since $\alpha$ is the root of a irreducible polynomial ...
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18 views

polynomials whose singular locus is irreducible/dimension zero

Do we understand the conditions under which the singular locus of a polynomial (the set of zeros of the polynomial and all its first order derivatives) is an irreducible variety ? Also, how about ...
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0answers
28 views

$A$ is matrix with integer entries and determinant 1. Effect on determinant if we reduce the entries modulo $k$

Suppose I have a $n\times n$ matrix $A$ with integer entries. Suppose $\det(A)=1$. I was thinking how determinant will change if I reduce the entries of $A$ modulo $k$ where $k$ is a positive ...
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37 views

Prove that $\left(\forall\ a \in A \right)\ aTe = eTa = e$

Let $\left(A,\ *, T\right)$ be a Ring and $e$ be its identity element. Is it possible to prove the following statement without using the regularity of $aTe$ in the underlying group $\left(A,\ *\right)...
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1answer
10 views

Clarification of summation term in a linear combination

Could someone clarify for me with the summation symbols are needed here? Aren't $\alpha_h$ and $\beta_k$ enough to construct every element of W?
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1answer
25 views

Set Ring and Algebraic Ring

Let $\mathscr{R}$ be a ring of sets and define set operations $\odot=\text{multiplication}$ and $\oplus=\text{addition}$ by $$E\odot F=E\cap F$$ and $$E\oplus F=E\triangle F$$ then $\mathscr{R}$ is ...
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1answer
16 views

Show that the ring of upper triangular 2x2 matrices is a direct sum of two of its modules

The definition I have for the direct sum of R-modules is the following: $\bigoplus_{i \in I} M_i = \big\{f \in \Pi_{i\in I}M_i : \ \#\{i \in I:\ f(i) \neq 0 \} < \infty \big\}$. Say k is some ...
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1answer
29 views

Show that $\operatorname{GL}_2 (\mathbb{F}_3)/\{\pm I_2\} \cong S_4$

Let $\mathbb{F}_3$ be a field with three elements and let $V = \mathbb{F}_3^2$. Let $\alpha,\beta,\gamma$ and $\delta$ be the four one-dimensional subspaces spanned by $\begin{bmatrix}1\\0 \end{...
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one's cell phone data is as unique as one's DNA to that person [on hold]

I have been been a digital forensics examiner for twenty years now. It is my position that a person's cell phone, especially the longer the person has owned and used the same cell phone, becomes as ...
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68 views

How would I use $x + 1= 0$ to conclude that $x + x =1$? [on hold]

In terms of fields Let $f = \{ 0,1,x\}$ , how would I go on to prove this using the given field axioms of closure, associativity of addition and multipication, commutativity of addition and ...
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1answer
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In proving G contains an element of order 15 if contains normal subgroups of orders 3 and 5, respectively, is $HK$ itself cyclic with order 15?

There is an answer here, but it is a "roadmap". group containing normal subgroups of orders $3$ and $5$ contains element of order $15$ There are answers here, but they are "roadmaps" too. If $G$ ...
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1answer
26 views

What is the definition of the least common multiple of elements of a group rather than their orders?

In an answer to my question here Relationship between order of $(x,y)$ in $G \times G'$ and order of disjoint permutations, which both involve LCMs?, it is said that Let $G$ be a group, $a\in G$...
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1answer
29 views

Is $NK=KN$ still even if only one of them is normal but both are still subgroups?

In this question Prove that the product $NK$ of two normal subgroups $N$ and $K$ of a group $G$ is a normal subgroup of $G$, and $NK=KN$., it is proved that $NK=KN$ if both $N$ and $K$ are normal ...
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1answer
44 views

Find number of invertible elements in $\mathbb{Z}[i]/(220+55i)\mathbb{Z}[i]$

I was able to find the factorization $220+55i=11*(2+i)*(2-i)*(4+i)$. Also know this famous question Quotient ring of Gaussian integers But how to apply it in this case? I'm confused, please help. ...
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1answer
39 views

Points of continuity in $ M_n (k) $ of minimal polynomials

The goal is to show that $\Gamma $ the set of points of continuity of $ M \mapsto \pi_M $ is $ \{ M \in M_n(k) , \chi_M = \pi_M \} $ . Where $ \pi_M $ is the minimal polynomial of $M$ and $ \chi_M $ ...
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1answer
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Relationship between order of $(x,y)$ in $G \times G'$ and order of disjoint permutations, which both involve LCMs?

The following is an exercise from Artin's Algebra: (Kiefer Sutherland's voice) Let x be an element of order r of a group G, and let y be an element of G' of order s. What is the order of $(x, y)$ ...
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13 views

How to compute $\operatorname{Ker}(\phi )$ and $(\phi )(25)$ for $(\phi ):Z\rightarrow Z_{7}$ such that $\phi(1)=4$? [duplicate]

I understand I must find some element that map to identity but what condition $\phi(1)=4$ is related ? and how to compute $(\phi )(25)$? I must find mapping from 25 to $Z_{7}$ isn't it ? is that 4 ?
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1answer
41 views

Ideals of $\Bbb{Z}_4[X]/(X^2+X+1)$

I want to find (maximal) ideals of $\Bbb{Q}[X]/(X^2)$ and $\Bbb{Z}_4[X]/(X^2+X+1)$. for the 1st one I can do this, because $(x)$ is a maximal and $(x)^2=0$. Now, I want to know how ideals (specially ...
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1answer
60 views

Is there a name for, or notable structure that uses, weird “distributive laws” such as $a\times(b+c)=b\times a+c\times a$?

Consider the following "multiplication" over "addition": $$ a \times (b + c) $$ The distributive law in common notion is the left distributive law: $$ a \times (b + c) = a \times b + a \times c $$ But ...
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1answer
38 views

Prove $[G:H]=[G':H']$ where $H$ and $H'$ are corresponding subgroups.

For a surjective homomorphism $\varphi: G \to G'$ where $\ker \varphi \subseteq H$, $H$ is a subgroup of $G$ that corresponds to $H'$, a subgroup of $G'$, the correspondence theorem implies $$\varphi(...
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1answer
27 views

Torsion elements of a group aren't necessarily a subgroup [duplicate]

In a question from an exam in an undergraduate group theory course, we were asked to prove or disprove that the set of all Torsion elements of a group is necessarily a subgroup. I knew that the set ...
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0answers
31 views

Ring theory:Ideals

Under the map $$\phi:R\rightarrow R/N$$ $$r\rightarrow r+N$$ where $R$ is a field and $N$ is an ideal of $R$. it is a well known facts that the image of an ideal ...
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32 views

Question from Algebra: Chapter 0 by Aluffi [duplicate]

Let K be a normal subgroup in finite group G. Assume |K| and [G:K] are relatively prime. Is K characteristic in G? I am self-studying through this book. Please only provide very small hints.
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1answer
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Prove that the quotient and remainder of division a and b in $\mathbb{Q}$[X] also belong to $\mathbb{Z}$[X].

Suppose a and b are polynomials with integer coefficients and b has a leading coefficient 1. Prove that the quotient and remainder of division a and b in $\mathbb{Q}$[X] also belong to $\mathbb{Z}$[X]....
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16 views

Arguing that $(A\cap B)/N = A/N \cap B/N $

Let $N\leq A, B\leq G$ where $N\trianglelefteq G$ so that $G/N$ is a group. With correspondence theorem, I am trying to show that the join and intersection of $A$ and $B$ has a unique correspondence ...
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1answer
27 views

Problem about index of proper nontrivial subgroup

Show that a finite simple group $G$ of order $\geq d!$ can not have a proper nontrivial subgroup of index $d$. Remark: I guess that the condition of this problem is a bit incorrect, namely we need to ...
2
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1answer
19 views

Counting irreducible polynomial of degree 3 over finite fields with certain restriction

I want help in the following counting problem: We know how many irreducible Monic polynomial of degree 3 are there in $\mathbb{F}_{q}[x]$. Now what I want to find is number of monic irreducible monic ...
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20 views

Give Dedekind cuts corresponding to the following [on hold]

Can anyone help me define the Dedekind cuts for these numbers? a) √3 b) fourth square of 2 c) 3 - √2
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0answers
21 views

Let f represent a set, how do I prove if this is a field or not? [on hold]

If I have a set {+- 1/8, +-1/4, 0, 1, +-1, +-2, +-4, +-8} with the usual addition. And multiplication, how do I find out or prove if this a field or not?
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1answer
31 views

Subgroups of symmetry Group

I'm reading Matrix Groups for Undergraduates by K.Tapp. In chapter $3$ he defines symmetry group as the group of all isometries of $\mathbb{R^n}$ that carry $X\subset\mathbb{R^n}$onto itself: $Symm(...