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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Semi lattice of groups!

Can someone please explain to me: How can I connect the structure of the Clifford semigroup with the structure of its maximal group image? Thanks!
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18 views

Describe group $(\mathbb{Z_5^*}, *_5)$ using Caylay table.

I am not quite sure what $\mathbb{Z}$ means, i mean, $\mathbb{Z}$ is a set of whole numbers, but that index 5 confuses me, d don't know what it means, and again, what $*_5$ means, i mean it is a ...
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44 views

non-zero divisors in a ring

I am asked to show the following: $ab$ is a non-zero divisor of $R$ if and only if $a$ and $b$ are both non-zero divisors of $R$. $\Rightarrow)$ Suppose $ab$ is a non-zero divisor of R. Then ...
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1answer
71 views

Find the subrepresentation of a cyclic group

This is a spin-off of this question: Show that representation $\rho$ can be divided I came across the problem of dividing representation $\rho$ of a cyclic group given as below: $$ g \longmapsto ...
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30 views

When is the ideal generated by 2 elements equal to the sum of the 2 ideals

Is it true in general that (a,b)=(a)+(b)? I would suppose that (a)+(b)$\subset$(a,b) and i believe the reverse containment should hold as well, i just can't seem to fit the pieces together.
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Why is $(A \otimes_\mathbb{Z} V_p )\bigcap R(G) = V_p$

I am reading the proof of brauers theorem in Serres book Linear Representations of finite groups and I have trouble understanding Lemma 5(page 75). Let $G$ be a group of order $g$. First let $g=p^nl$ ...
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1answer
33 views

If there are nonzero elements $a$ and $b$ in $A$ such that $(a+b)^2 = a^2 + b^2$, then $A$ has characteristic 2.

Let $A$ be a finite integral domain. If there are nonzero elements $a$ and $b$ in $A$ such that $(a+b)^2 = a^2 + b^2$, then $A$ has characteristic 2. I was thinking, if $(a+b)^2 = a^2 + 2ab + b^2 = ...
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27 views

Let $n>0$ and $m>0$ be integers, and let $c = \gcd(n,m).$ Show that $\gcd\left(\frac{n}{c},\frac{m}{c}\right) = 1.$

Let $n>0$ and $m>0$ be integers, and let $c = \gcd(n,m).$ Show that $$\gcd\left(\frac{n}{c},\frac{m}{c}\right) = 1.$$ I attempted using the idea that we know from definition of $gcd$ ...
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15 views

Laurent Ideal whose Intersection with Polynomial Ring Requires More Generators

I want to find an ideal $I\subseteq \mathbb Q[x^{\pm 1}, y^{\pm 1}, z^{\pm 1}]$ which requires fewer generators than the affine ideal $I\cap \mathbb Q[x, y, z]$. I tried finding a principal ideal $I$ ...
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37 views

Show that representation $\rho$ can be divided

let $\rho$ be a representation of a cyclic group of order 6 defined by the following relationship: $$ 1 \longmapsto \begin{pmatrix} 1 & -1 \\ 1 & 0 \\ \end{pmatrix} $$ ...
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21 views

How Do I Apply the Cantor-Zassenhaus algorithm to $\mathbb{F}_2$?

Recently, I've been trying to implement the Cantor-Zassenhaus algorithm in C++ over $\mathbb{F}_2$. According to this lecture, the algorithm is basically: Input is polynomial $f\in\mathbb{F}_q$ with ...
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25 views

If F is a simple module, is $F^n $ semi simple?

If F is a simple module, is $F^n $ semi simple? I'm assuming this is true by the classification theorem
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2answers
49 views

Prove R has no nonzero nilpotents if an ideal A has no nonzero nilpotents and R/A has no nonzero nilpotents. [closed]

I want to prove the following implication: Let $R$ be a ring and $A$ an ideal of $R$. Suppose that $A$ has no nonzero nilpotents and $R/A$ has no nonzero nilpotents. Prove that $R$ has no nonzero ...
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1answer
41 views

If $G = G_{\alpha} \rtimes R$ and $R$ is non-abelian of order $27$ and exponent $3$, then $G_{\alpha} \cong C_2\times C_2, D_3$ or $D_4$

Suppose that $G$ is a transitive permutation group and let $R$ be a regular normal subgroup which is isomorphic to the non-abelian group of order $27$ and exponent $3$. Then $G = R \rtimes ...
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31 views

Valuation ring between $F$ and $\mathcal O_F$

Let $(F,v)$ be a complete discrete valuation field (normalized) with ring of integers $\mathcal O_F$. Why cannot exist a valuation ring $A$ of $F$ such that $F\supsetneq A\supsetneq\mathcal O_F$ ...
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30 views

Regular representations of $Z_2 \times Z_2$ and $Z_4$ and their decomposition

I've tried to look for info about this but failed. First off, what does a regular representation of $Z_4$ and $Z_2 \times Z_2$ look like? I know that a regular representation for a group $G$ is a ...
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32 views

The natural numbers form a distributive lattice under gcd and lcm. In arbitrary gcd domains, does gcd distribute over lcm?

Basically what it says in the title. If $A$ is a $\operatorname{gcd}$ domain, for any $x, y, z \in A$, does this identity hold? $$\operatorname{gcd}(x, \operatorname{lcm}(y,z)) = ...
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3answers
45 views

Suggestions for readings; Elliptic curves over function fields

I would love to know some good refercences about Elliptic curves over function fields. Especially in view with Mordell-Weil's Theorem. I am already familiar with the main proof of Mordell's theorem in ...
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1answer
25 views

Symmetric Polynomial in roots is in $F[X]$

I recently came across the following claim. Let $F$ be a field of characteristic $0$. Let $f\in F[X]$ have roots $y_1, \ldots , y_d$ in the algebraic closure of $F$. Define $$ g_h = \prod_{1\le ...
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69 views

Notation: $\mathbb{Z}[\sqrt{-5}]$

Show that the elements 2,3, and $1 \pm \sqrt{-5}$ are irreducible elements of $\mathbb{Z}[\sqrt{-5}]$. I have never seen this notation before. From another post I am interpreting this to mean the ...
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2answers
49 views

Well-defined function

Prove or disprove that the following functions are well-defined for all $m \geq 2$: $$ f : \mathbb Z_m \rightarrow \mathbb Z_m, \overline x \mapsto \overline{x^2} $$ $$ g : \mathbb Z_m \rightarrow ...
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3answers
45 views

Sum of all elements in congruence class modulo n

With $+$ defined as $[a]+[b]=[a+b]$, show that $[0]+[1]+\cdots+[n-1]$ is equal to either $[0]$ or $[n/2]$ in $\Bbb Z_n$. How do I go about proving this? I have managed to get $[(n^2-n)/2]$ using the ...
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42 views

Prove that the sum of ideals of a ring A equals A and its intersection is zero.

I've been looking at a couple of ring theory exercises and there's this one I don't know how to do it. It goes like this. $A$ is a commutative unital ring, and $e$ an element of $A$, $e \neq ...
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34 views

Origin of the name “Cauchy sequence” in abstract algebra

I know the origin in analysis as it is derived from our good old chap Cauchy himself. However I have read about Cauchy sequences in algebra, I'll use groups for this one, let $G$ be a group and ...
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1answer
44 views

The number of elements in the special linear group over the finite field $\mathbb{Z}/p$ [closed]

I have $SL_{2}\{\mathbb{Z}/p\}$ for $p$ prime and $\mathbb{Z}$ integers. How do I show that this is a subgroup of $GL_{2}\{\mathbb{Z}/p\}$ and find the number of elements in $SL_{2}\{\mathbb{Z}/p\}$?
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1answer
22 views

Normalizer probtlem for finite nilpotent groups

Lemma- Suppose $P$ is a $p$-group contained in $G$ and $u\in N_U(G)$ where $U=U(\Bbb{Z}G)$. Then there exist $y\in G$ such that $u^{-1}gu=y^{-1}gy\ \forall\ g\in P$. We use this lemma to prove ...
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1answer
30 views

Integral element in $\mathbb{Q}(\sqrt{2})$ over $\mathbb{Z}$ has form $m+n\sqrt{2}$ [duplicate]

I have studied about integral element. And I am not clear about an example. Could you please help me explain about it? Let R is a subring of S. $s \in S$ is integral if $s$ is a zero of some monic ...
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21 views

If $G$ is of order $n$ and $k$ is prime to $n$ then $g(x)=x^k$ is homomorphism

Let $G$ be a group of order $n$ and let $k$ be prime to $n$, show that $g(x) = x^k$ is one-to-one. I started trying to prove this, and said: If $g(x) = g(y)$ for some $x,y \in G$ then $x^k = ...
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51 views

Prove that a group G is finitely generated if and only if there is a surjective homomorphism $F(\{1,..,n \}) \to G$

This if from Aluffi's Algebra: Chapter 0. There is an another definition of subgroup generated by a subset. Here it is: Let $G$ be a group and $A$ its subset. By universal property of a free ...
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1answer
20 views

Why must the image of the identity of nonzero ring $A$ must be a zero divisor of the codomain?

Lets say I have a ring homomorphism $f$ which maps from the nonzero rings $A$ to $B$, both of which have identities, but it maps the identity of $A$ to something other than the identity of $B$. Why is ...
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1answer
36 views

Zero divisors within the ring of dual numbers

Let $\mathbb{R}(\epsilon) = \{ a + b \epsilon : a,b \in \mathbb{R} \}, $ where $(a + b\epsilon) + (c + d\epsilon) = (a+c) + (b+d)\epsilon$ and $(a+ b\epsilon) \cdot (c + d\epsilon) = ac + (ad + ...
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1answer
23 views

Surjectivity and the non-existence of maps.

This question comes from Jacobson's Basic Algebra. It asks: Show that $S \overset{\alpha}{\to} T$ is surjective iff there exist no maps $\beta_1,\beta_2$ of $T$ into a set $U$ such that $\beta_1 ...
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1answer
21 views

Characterize units in formal power series $R[x]$

Suppose R is a commutative ring with unity. Define $R[x]$ as "formal power series in the variable $x$ with coefficients from R". These are the infinite sums of the form $ \sum_{n=0}^\infty ...
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80 views

Is the element $(0,0,0)\in\mathbb{R}^3$ a divisor of zero?

(I'm assuming that $\mathbb{R}^3=\mathbb{R}\times\mathbb{R}\times\mathbb{R}$) In my assignment, I'm told to prove that exactly one of the following can be true for an element $(x,y,z)\in\mathbb{R}^3$ ...
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1answer
22 views

Why is the set of units of integer quaternions isomorphic to the quaternion group of order 8?

Let's say that I've got a ring $V$ of integer quaternions of the form $\mathbb{Z} + \mathbb{Z}i + \mathbb{Z}j + \mathbb{Z}k$. Now assume that there exists an element $a = a_1 + a_2i + a_3j + a_4k$ ...
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27 views

Show that for $p \neq 2$ not every element in $\mathbb{Z}/p\mathbb{Z}$ is a square.

Show that for $p\neq2$, not every element in $\mathbb{Z}/p\mathbb{Z}$ is a square of an element in $\mathbb{Z}/p\mathbb{Z}$. (Hint: $1^2=(p-1)^2=1$. Deduce the desired conclusion by counting). So far ...
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Why does $ab=ba=1$ imply ${a_1}^2 + {a_2}^2 + {a_3}^2 + {a_4}^2 = 1$?

Let's say that I've got a group $V$ of integer quaternions of the form $\mathbb{Z} + \mathbb{Z}i + \mathbb{Z}j + \mathbb{Z}k$. Now assume that there exists an element $a = a_1 + a_2i + a_3j + a_4k$ ...
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How to compute? [duplicate]

Let $F=\mathbb Q[x]/\langle x^n +1\rangle$ and $n$ is an integer. Define $|f| = \sqrt{a_0 ^2 + a_1 ^2 +\cdots + a_{n-1}^2}$, where $f = a_0 + a_1 x + \cdots + a_{n-1}x^{n-1}$. If $f \in F$ satisfies ...
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76 views

List all elements of $\mathbb{Z}_5[x]/\langle x^2+3x+1\rangle$

List all elements of $\mathbb{Z}_5[x]/\langle x^2+3x+1\rangle$ Is there a simple way to solve these kind of questions (only using pen and paper)? If I understand it correctly, I can multiply ...
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1answer
40 views

Absolute Galois group $\text{Gal}(\overline{K}/K)$ of any number field $K$ has a non-open subgroup of any prime index $p$?

Let $K$ be a number field, and let $p$ be a prime number. Does $G = \text{Gal}(\overline{K}/K)$ necessarily have a subgroup of index $p$ that is not open?
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24 views

$G/\lambda_2$ is nilpotent?

Let $G=\lambda_0 \geq \lambda_1 \geq \lambda_2 \ldots$ where $\lambda_i=[\lambda_{i-1} , G]$ be the lower central series. Then is it true that $G/\lambda_2$ is nilpotent? If true how can I prove this. ...
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258 views

Eigenvector and eigenvalue for exponential matrix

$X$ is a matrix. Let $v$ be an eigenvector of $e^{X}$ with corresponding eigenvalue $a$. Show that $v$ is also an eigenvector of $e^{X}$ with eigenvalue $e^{a}$ If $X$ is diagonalizable, then we can ...
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1answer
29 views

Saturation of a multiplicatively closed subset

Exercise 3.7 of Atiyah-MacDonald asks the reader: if $A$ is a commutative ring and $\mathfrak{a} \triangleleft A$ an ideal, find the saturation of $1 + \mathfrak{a}$. Previously we have shown that ...
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1answer
51 views

What are the units in $\mathbb{Z}/n\mathbb{Z}$ in general?

What are the units in $\mathbb{Z}/n\mathbb{Z}$ ($n$ is any positive integer) in general? I figured it should a group under multiplication mod $n$, but was wondering if there is any more specific way ...
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41 views

Finding the radical of $\mathfrak{gl}(2,\mathbb{C})$ [duplicate]

I am taking a Lie algebras course as a prerequisite to study Lie groups. The idea of a radical of a Lie algebra (maximal solvable ideal) has been defined in class but no other statements or theorems ...
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16 views

Neutral element

Let $G = (M, \circ )$ be a groupoid and let $2^G = (2^M, \circ_K)$ a groupoid ( $\circ_K$ is the product of group subsets). I want to show that $2^G$ has a neutral element iff $G$ has a neutral ...
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1answer
43 views

Prove that $\text{im}(T^t)=(\ker T)^0$

Let $V$ be a finite dimensional vector space over $F$ and $T:V \rightarrow W$ linear transformation. Prove that $\text{im}(T^t)=(\ker T)^0$ where $T^t$ is the dual/transpose of $T$ and $(\ker T)^0$ ...
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1answer
42 views

Can anyone explain how primitive roots work?

Right now I'm studying out of Audrey Terras' book Fourier Analysis on Finite Groups and Applications and we're on the section where we're talking about $(\mathbb{Z}/n\mathbb{Z})^*$ and when this group ...
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1answer
54 views

Alternative proof that every subgroup of a cyclic group is cyclic

I've seen the 'standard' proof where we essentially construct a generator and use the division method. I was thinking through the problem attempting to do this using isomorphism's and was looking for ...
4
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50 views

Prove that there exist a sylow subgroup of $G$ which is fixed by $\alpha$.

Let $G$ be a finite group and $\alpha\in Aut(G)$ such that $\alpha$ restricted to any sylow subgroup of $G$ equals the restriction of some inner automorphism of $G$ and $(o(\alpha),o(G))=1$, then is ...