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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find all subgroups of G where: $0 \ne r \in \Re$ $G = <r>$

I need to find all subgroups of G where: $G \lt \Re$ $0 \ne r \in \Re$ $G = <r>$ $\Re$ is the group of real numbers and G is a subgroup. Edit : the operation is + I tried thinking about ...
5
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1answer
40 views

Given a group of order $p^nq^2$ for two odd primes, prove that the commutator is a p group.

Given a group of order $p^nq^2$ for two odd primes $p > q$, prove that the commutator is a p group. To solve this question I need to prove that the commutator can't be of the orders $p^iq$, ...
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20 views

proofing $Z(G)=\langle [x,u]\rangle$ if $M=C_G(u)$ is maximal subgroup

Let $G$ be non-abelian finite p-group, $p$ is odd, with cyclic center and $u\in G$ be of order $p$ if $M=C_G(u)$ (centralizer of $u$) be a maximal subgroup and $Z(G)\le M$ for $x\in G\setminus M$ how ...
1
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1answer
23 views

eigenvalues of cycle graph and its complement graph

I am trying to find the eigenvalue of cycle graph and its complement. How to simplify.Suppose $\omega^{1}+\omega^{n-1}=2\cos (2\pi/n) $, then, $\omega^{\frac{n-1}{2}}+\omega^{\frac{n+1}{2}}=\ ?$ Is ...
6
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53 views

Explicit examples of (co)limit arguments in other fields

Over the past weeks, I have noticed that high level lecture notes in subjects like algebraic geometry, algebra, and algebraic topology often sketch proofs in the following form: Proof sketch ...
2
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0answers
24 views

calculation $p$-Fitting subgroup

Let $ G $ be a finite soluble group and $ A $ is the unique minimal normal subgroup of $ G $ that $ \vert A \vert = p^{a} $, $ p $ is prime. Let $ N =Fit(G) $, then $ N = O_{p}(G) $. Suppose $ F/N = ...
10
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4answers
1k views

Why is it necessary for a ring to have multiplicative identity?

I have read earlier that in a ring $(R,+,.)$ the following needs to hold: $(R,+)$ is an abelian group multiplication is associative and closed left and right distribution laws hold. However, I ...
2
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0answers
56 views

A finite von Neumann regular ring is unital and has $ab = 1$ if $ba = 1$

Let $R$ be a finite ring satisfying for any $x \in R$ there exists $y \in R$ with $xyx = x$. Show that $R$ is unital and that if $ab = 1$, then $ba = 1$. Thoughts so far: If I can show that the ...
1
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1answer
40 views

$x$ and $g$ are elements of the group $G$, show that the order of $x$ is equal to the order of $g^{-1} xg$.

If $x$ and $g$ are elements of the group $G$, prove that $|x| = | g^{-1} xg|$. Deduce that $|ab| = |ba|$ for all $a,b \in G$. attempt: Let $|x| = n$ be the order of $x$ and $| g^{-1} xg| = m$ be the ...
2
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2answers
45 views

Existence of integer solution of $a^2 -17b^2 = $ any constant

When checking whether if $9-\sqrt{17}$ in the ring $\{a+b\sqrt17: a,b \in \mathbb{Z}\}$ is a prime. Suppose $$\alpha\cdot \beta = 9-\sqrt{17},$$ using norm argument $$N(\alpha)N(\beta) = ...
3
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1answer
55 views

Is there a nice expression for $f(x) = (1+x)(1+x^2)(1+x^3)\cdots$

While I was solving a problem, I stumbled upon this function $$f(x) = (1+x)(1+x^2)(1+x^3)\cdots$$ I tried to write out the first few products but I couldn't recognize any meaningful pattern. Is there ...
0
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1answer
34 views

Is a finite monoid with left cancellation property always a group?

I need to answer and show if a Monoid with left cancellation property always a group. I managed to show that it is correct when cancellation property holds for both left and right (that was part a of ...
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2answers
28 views

Understanding the connection of the roots of an irreducible polynomial and a basis for field extensions

Let $\alpha,\beta,\gamma \in E$ be the roots of an IRREDUCIBLE polynomial $p(x)\in Q[x]$ (where E/Q is an extension field. Can I use these roots to construct a basis for E over Q? Why?
5
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38 views

For finitely generated free abelian groups $A,B$ if there is an onto homomorphism $A \to B$, then $\operatorname{rank}(A) \geq \operatorname{rank}(B)$

$\newcommand{\rank}{\operatorname{rank}}$For two finitely generated, free abelian groups $A,B$ prove that if there is an onto homomorphism $A \rightarrow B$, then $\rank(A) \geq \rank(B)$ Assume that ...
2
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1answer
28 views

How to create a ring in MAGMA with relations?

I'm using MAGMA221 and would like to create a ring over $GF(2)$ with respect to a list of relations. Here's what I have so far: $\mathtt{Z:=GF(2);} \\\mathtt{P<x,y,z>:=PolynomialRing(Z,3);}$ ...
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28 views

Prove that the center of a group with $385$ elements has an element of order 7. [duplicate]

Prove that the center of a group with $385$ elements has an element of order 7. By Cauchy's theorem I know that if I prove that $Z(G)$'s order is divisable by 7, we're done. So now I need to rule out ...
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0answers
32 views

When the fibers of $A \to A[T]/(h(T))$ are geometrically regular?

Let $A$ be an affine commutative noetherian domain over a characteristic zero field $K$, $T$ an indeterminate, $h=h(T) \in A[T]$ (not necessarily monic), $B=A[T]/(h)$, and assume that $(h)$ is a prime ...
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1answer
35 views

Empty set - group [duplicate]

I started a course of algebra this morning, and the teacher explained the structure of a group. He explicitly explained a group has to be empty. Someone can explain to me why it is a necessity for a ...
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2answers
53 views

Between complex numbers and quaternions?

Complex numbers are $a+ b i $; Quaternions are $a + b i + c j + d k $. So, do there exist numbers like $a + b i + c j$? Here $a$, $b$, $c$, $d$ are all real. Did Hamilton consider such a case?
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1answer
42 views

Let $V=\{i\in \mathbb{Z}: 0\leq i< 2^n\}$. Define vector addition and scalar multiplication on $V$ to turn it into a vector space over $GF(2)$.

Let $V=\{i\in \mathbb{Z}: 0\leq i< 2^n\}$ for some $n\in \mathbb{N}$. Define vector addition and scalar multiplication on $V$ in such a way as to turn it into a vector space over the field ...
2
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1answer
25 views

Evolution operator always in $SU(n)$?

Think about the evolution operator $U$ in Quantum Mechanics for finite dimensional systems. Then this operator satisfies an equation $$U'(t) = -iHU(t)..$$ Here, I assume that $H$ is ...
1
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1answer
39 views

How to prove that O(Ng) | O(g)

I have this exercise: Let $G$ be a group and $N$ a normal subgroup of $G$. Show that for all $Ng\in G/N$, $$o(Ng)\mid o(g).$$ For now, without using the canonic homomorphism $\tau ...
2
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2answers
164 views

Group Action as permutations

I'm trying to study on Group actions. the paper says(if I understand) that if I have Set $S$ and an action $\alpha$: $G \times S \rightarrow S$ . then the action may be viewed as permutation by $x ...
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1answer
36 views

Are subgroups of order $p^{n-1}$ maximal?

Let $G$ be finite p-group of order $p^n$, I know all maximal subgroups of order $p^{n-1}$ Is it right to say all subgroup of order $p^{n-1}$ are maximal subgroups? If not, what property $G$ must have ...
0
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0answers
30 views

Rings leading to AKS primality test

Given number n, define ring $R = \Bbb Z_n[x]/(x^r −1)$ for a carefully chosen number $r$ ($r$ is much smaller than $n$; of the order of square of the number of digits in $n$) An element of $R$ is a ...
0
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2answers
47 views

Group of an order 385

Let $G$ be a group of order $385$, proof that $Z(G)~~ (cent(g))$ contains object of order $7$. I used sylow theorm and realized that there are $1$ sylow-$11$ sub-group which is normal in $G$ and also ...
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18 views

Enumerating functions modulo action on both the domain and codomain.

Let $Hom(A,B)$ be the set of functions from a finite set $A$ to a finite set $B$ and let $G_A \leq S_A$, $G_B \leq S_B$ be a subgroups of the permutation groups of $A$ and $B$. For $f,g \in Hom(A,B)$ ...
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0answers
28 views

Factor rings $R/R$ and $R/0$

Let $R$ be a ring. I want to describe the factor rings $R/R$ and $R/0$. So $R/R = \{[r]| r+R, \forall r\in R \}$ and since $r+R=R$, we get $R/ R =\{[0]\}$. And for $R/0 = \{[r]| r+0,\forall r\in ...
2
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1answer
21 views

Clarification on wording of a problem in Hungerford's Algebra

I'm currently working on problem 11 in section 1.1 of Hungerford's graduate text, which is to show 5 conditions of a group are equivalent. However, I'm not exactly sure what the last condition means: ...
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24 views

Elements that aren't left zero divisors are invertible for certain group algebra

Let $G$ be a finite group and $F$ a finite field with co-prime orders. Show that in the group algebra $F[G]$, if $x \in F[G]$ is not a left zero divisor then it is invertible. Thoughts so far: By ...
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0answers
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There exists an $f\in \Bbb F[x]$ such that $I=\{fg|g\in \Bbb F[x]\}$.

$I\trianglelefteq \Bbb F[x]$. I want to prove that there exists an $f\in \Bbb F[x]$ such that $I=\{fg|g\in \Bbb F[x]\}$. I guess this means that I am meant to show that we have closure from the ring ...
0
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1answer
55 views

Define multiplication as $(a,b) \boxdot (c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$ where $a,b,c,d,r\in \mathbb{R}$ and $0\neq s\in \mathbb{R}$.

Define multiplication as $(a,b) \boxdot (c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$ where $a,b,c,d,r\in \mathbb{R}$ and $0\neq s\in \mathbb{R}$. The multiplicative inverse is $(1,0)$. I need to show that ...
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33 views

Elementry of analysis [on hold]

I don't understand this question, could you please define the question for me and tell me what should I do? Consider a particular device capable of arithmetic computation. Consider the operation ...
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0answers
39 views

The number of all groups with n elements? [on hold]

Let $n$ be a positive integer number. How many groups of $n$ elements (which are not isomorphic) ?
10
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1answer
132 views

Does $f(x) \in \mathbb{Z}[x]$ irreducible, imply $f(2x)$ also irreducible?

It is well known that $f(x+a) \in \mathbb{Z}[x]$ is irreducible when $f(x)$ is irreducible. I was wondering whether $f(2x)$ and in general whether $f(nx)$ is irreducible as well. Though it is ...
2
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2answers
40 views

How does uniqueness of the additive inverse imply that $-(ax) = (-a)x$?

How does uniqueness of the additive inverse imply that $-(ax) = (-a)x$? In my title, I should be clear that the additive inverse should be unique. But how does this help? I dont even get why ...
0
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1answer
41 views

Left remainder when dividing by $x-b$

Give a polynomial $p(x) = a_0 + a_1 x + ... a_n x^n \in \mathcal R[x]$ ($\mathcal R $ is any ring with unity), the book says when dividing $p(x)$ by $x-b \quad (b\in \mathcal R)$, the left remainder ...
3
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1answer
87 views

Equivalent conditions for an ideal to be zero-dimensional.

For an ideal $I \subset \mathbb{C} [x_1, ... , x_n]$ show that dim$_{\mathbb{C}}R/I$ is finite iff $I$ is contained in only finitely many maximal ideals. Thoughts so far: I'm not sure how to get ...
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Suppose $ S = HF $ for $ H \leq S $ and $ H \cap F = 1 $. If $ N \leq H $, show $ N \leq Z(S) $.

Let $ G $ is a soluble group and $ N $ minimal normal subgroup of $ G $. Let $ F = Fit(G) $. Suppose $ S = HF $ for $ H \leq S $ and $ H \cap F = 1 $. If $ N \leq H $, show $ N \leq Z(S) $. I show ...
2
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1answer
32 views

Linear algebra: generalize from characteristic $0$ a problem about polynomial coefficients.

Let $K$ be a field, and let $F$ be a subfield of $K$. Assume that $F$ is infinite. Let $p(x)$ be a polynomial in one variable with coefficients in $K$, and suppose that $p(a) \in F$ whenever $a \in ...
0
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1answer
19 views

Linear transformations for fixing the line $y = 0$

The professor says that the subgroup for "stabilizing" the line $y = 0$ is $$A = \begin{bmatrix} a & c \\ 0 & d \end{bmatrix}$$ because in order to fix the first basis vector, $b = ...
5
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1answer
33 views

Classification of all conjugacy classes of $GL_2(\mathbb{R})$, $GL_2(\mathbb{Q})$.

Give a classification of all conjugacy classes in the following groups. $GL_2(\mathbb{R})$ $GL_2(\mathbb{Q})$ My progress so far. If the characteristic polynomial splits, the matrix ...
1
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1answer
14 views

prove unique minimal normal subgroup of soluble group by $ P_{G}(M) > M $

Let $ G $ be a soluble group. If $ P_{G}(M) = \langle y\in G | \langle y \rangle M = M\langle y \rangle \rangle > M $for any subgroup $ M $ of prime power index in $ G $, then every chief factor ...
4
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1answer
23 views
+50

$\mathbb{C}\{X\}^\chi$ a $G$-stable subspace of $\mathbb{C}\{X\}$?

Let $\mathbb{F}$ be a finite field with $q$ elements, and let $G = SL_2(\mathbb{F})$. The group $G$ acts on the set $X := \mathbb{F}^2 \setminus \{0\}$, the complement of the origin. For any group ...
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2answers
42 views

$(a,b) \mathbin\# (c,d)=(a+c,b+d)$ and $(a,b) \mathbin\&(c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$. Multiplicative inverse?

Let $r\in \mathbb{R}$ and let $0\neq s \in \mathbb{R}$. Define operations $\#$ and $\&$ on $\mathbb{R}$ x $\mathbb{R}$ by $(a,b) \mathbin\#(c,d)=(a+c,b+d)$ and $(a,b) ...
4
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0answers
57 views

A quick Galois Theory question

Let $\mathbb{F}_q$ be a finite field of order $q$ where $q$ is a prime power. For any $d \in \mathbb{N},$ we have an inclusion $\mathbb{F}_{q^d} \subseteq \overline{\mathbb{F}}_q.$ Both ...
2
votes
0answers
47 views

Find a polynomial equation satisfied by $\phi$

I'm solving the following exercise from my class notes: Let $A=k[x^2,y^2]$ and let $M=k[x^2,xy,y^2]$ ($k$ a field). Show that $M$ is an $A$-module. Define $ \phi :M \to M$ by $\phi(m)=xym$. Find ...
5
votes
2answers
85 views

Number of matrices $A \in M_n(\mathbb{F}_q)$ where $A^2 = 0$.

What is the number of matrices $A \in M_n(\mathbb{F}_q)$ for which $A^2 = 0$ (as a function of $n$ and $q$)?
4
votes
1answer
59 views

Is there a “unique factorization theorem” for finite groups?

Sometimes it is difficult for me to understand what a group seems like. For example, the dihedral group $D_5$ is easy to visualise when I think it of as a "product" of two cyclic groups $C_2$ and ...
3
votes
4answers
45 views

Prove that $R$ is an integral domain $\Leftrightarrow$ $R[x]$ is an integral domain

Here is an exercise(p.129, ex.1.15) from Algebra: Chapter 0 by P.Aluffi. Prove that $R[x]$ is an integral domain if and only if $R$ is an integral domain. The implication part makes no problems, ...