0
votes
0answers
11 views

General linear group/special linear group, isomorphic to R^(*)

Let $GL(n,\mathbb{R})$ be the group of invertible $n \times n$ real matrices, and let $SL(n,\mathbb{R})$ be the group of $n \times n$ real matrices of determinant $1$. And $\mathbb{R}^*$ be the group ...
-4
votes
0answers
31 views

Abstract Algebra: Prove that every field has only trivial ideals [on hold]

Prove that every field has only trivial ideals (that is, {0} and the field itself)
1
vote
3answers
44 views

Prove that if $G$ is a group and $H$ is a subgroup of $G$ generated by all elements of order $N$ in $G$, then $H$ is a normal subgroup of $G$.

I've tried proving that $ghg^{-1}\in H$ ($\forall g \in G$), but I don't see how the special property of $H$ guarantees this. Any insight? I've turned away from it to work on other things, and it's ...
0
votes
1answer
20 views

Proving $\dim(\ker( p (T) ) ) = n\cdot d$ where $n$ is a positive integer and $d$ is the degree of the polynomial.

A few more details: $T$ is $T : V -> V$ for some space $V$. Also, the polynomial $p$ is irreducible where $d \ge 1 $. What I've done so far was to restrict the transformation to the invariant ...
1
vote
2answers
25 views

Normal subgroup, quotient group, isomorphism.

Let $R^{*}$ be the group of nonzero real numbers under multiplication and let $R^{+}$ be the group of positive numbers under multiplication. Prove (a) $\{-1,1\}$ is a normal subgroup of $R^{*}$. (b) ...
-1
votes
1answer
21 views

A question of cosets

Show that if $H$ is a subgroup of index $2$ in finite group $G$, then every left coset of $H$ is also a right coset $H$. Thanks for all your helps. This is the last question of my homework :)
0
votes
2answers
47 views

Abelian isomorphic groups

Prove that if G and G' are isomorphic groups and G is abelian, then G' is abelian, too. I'll happy if you help me with this..
-1
votes
2answers
54 views

A question in Isomorphism

Let G be a cyclic group. Soppose G and G' are isomorphic groups. Show that G' is also cyclic. Can Someone Solve this pleaase? I have an exam 2 hours later!
0
votes
0answers
40 views

How can $\{ a+b \sqrt d : a,b \in \mathbb Z \}$ be a subset of $\mathbb C$?

I got this question on one of my homeworks. However as far as I am aware, $R \subseteq \mathbb C$ only when $ d= -1$. Am I overlooking something that makes it possible in the general case, or can ...
2
votes
1answer
50 views

$\mathbb{C} [G] \longrightarrow \prod_{\rho} \text{End}(V_{\rho})$ an intertwining isomorphism

Consider the vector space $\mathbb C[G]$ of functions $f: G \longrightarrow \mathbb{C}$ where $G$ is a finite group, or equivalently a vector space of all formal linear combinations of elements of $G$ ...
2
votes
1answer
31 views

Suppose that $G$ is a group of order 5 and let $\varphi: \mathbb{Z_{30}} \rightarrow G$ Determine the kernal of $\varphi$

Suppose that $G$ is a group of order 5 and let $\varphi: \mathbb{Z_{30}} \rightarrow G$ be and epimorphism. Determine the kernel of $\varphi$ Since $\ker\varphi \unlhd \mathbb{Z_{30}}$ (theorem) then ...
6
votes
4answers
114 views

If a polynomial $g$ divides $f$ and $f'$, then $g^2$ divides $f$?

Here's a homework problem from Artin's Algebra that I'm having a lot of trouble with Let $f(x) \in F[x]$ (where $F$ is a field of characteristic $0$). If $g$ is an irreducible polynomial that is ...
0
votes
1answer
20 views

Constructing nontrivial $\mathbb{Z}$-bilinear map from $\mathbb{R} \times (\mathbb{R} / \mathbb{Z})$

I'm trying to show $\mathbb{R} \otimes_\mathbb{Z} (\mathbb{R} / \mathbb{Z})$ is nontrivial, where my tensor product is defined using the universal property. This problem can be easily reduced ...
0
votes
1answer
22 views

tables of cyclic subgroups and conjugates

$G = S_5$, I need to construct tables for $H$ and $aHa^{-1}$ ($H =$ cyclic subgroup $(142)(35),$ and $a = (2354) \in G$) and see what can be inferred. In my attempt $H$ = $\{(142)(35), (124)(35), ...
0
votes
1answer
22 views

Showing that no non-identity element of $G/F_g$ has finite order where $G$ abelian, $F_g$ the set of elements of G that have finite order

Let $G$ be an abelian group and $F_g$ the set of elements of $G$ that have finite order. Show $F\trianglelefteq G$ and no non identity element of $G/F_g$ has finite order. $G$ is abelian $\implies ...
0
votes
1answer
25 views

Lie bracket of vector fields on $R^2$

Compute the Lie bracket$$\Big[-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y},\frac{\partial}{\partial x}\Big]$$ on $R^2$ Can you help me please?
2
votes
2answers
71 views

An application of Sylow theorems in p-groups!

If $G$ is a finite group of order $p^{n}$ (which $p$ is a prime number) and have only one subgroup of order $p^{n-1}$ ,namely $H$ ,then $G$ is cyclic ! My "proof" is as follows: suppose $$x\in G-H$$ ...
1
vote
3answers
28 views

Show that if $m\in M_n$ and $k \in \Bbb Z_n$ then $mk\in M_n$.

Let $M_n = \{a\in \Bbb Z_n\mid \text{ there exists a non-zero integer $k$ with the property that }a^k\equiv 0 \pmod n \}.$ Show that if $m \in M_n$ and $k \in \Bbb Z_n$, then $mk\in M_n$. For which ...
0
votes
0answers
14 views

Finding characteristic roots and characteristic vectors

V is a two-dimensional vector space over the field of real numbers, with a basis $v_1, v_2$. Find the characteristic roots and corresponding characteristic vectors for T defined by $v_1(T) = v_1 + ...
0
votes
1answer
64 views

Field of algebraic numbers over $\mathbb{Q}$

Let $F$ be the field of algebraic numbers over $\mathbb{Q}$. I do remember that this means $F$ is a field extension of rationals over $\mathbb{Q}$. How do I show that the field extension of $F$ is ...
2
votes
1answer
21 views

Counterexample to “A/I is Artinian, when I is the annihilator of Artinian A-module”.

Let M be an Artinian A-module and let I be the annihilator of M in A. Is A/I necessarily an Artinian ring? I believe the answer is no since this comes off of a similar result regarding Noetherian ...
1
vote
1answer
27 views

Show that if $N$ is a normal subgroup of $G$ which contains all commuters then $G/N$ is abelian.

I am working on my proof for class and I was wondering if this look ok? Let $N$ be a normal subgroup of $G$ we want to show that $G/N$ is abelian, or $(aN)(aN) = abN = baN = (bN)(aN)$. Since $N$ ...
4
votes
2answers
90 views

Is $i \in \mathbb{Q}[\sqrt[4]{-2}]$?

I have a homework question from Artin's Algebra that asks Is $i \in \mathbb{Q}[\sqrt[4]{-2}]$? I suspect that this is not true because $i \sqrt{2} \in \mathbb{Q}[\sqrt[4]{-2}]$ and $\sqrt{2}$ is ...
1
vote
1answer
35 views

Isomorphism preserves exactness

Let $R$ be a commutative ring with unity. Let $A_i$ be an R-module for every $i$. Consider a sequence of modules $$\xrightarrow{\delta_{i-1}}A_{i-1}\xrightarrow{\delta_{i}} ...
0
votes
0answers
28 views

Parity check matrix operations

Let $C_1$ and $C_2$ be linear codes of the same length over the finite field $F$, and let $H_1$ and $H_2$ be parity-check matrices of $C_1$ and $C_2$ respectively. Define $C_3$ as the code $C_3 = ...
2
votes
1answer
43 views

exercise of Matsumura

my question is about this exercise of Matsumura: in the proof hint we use is this obvious? or e should define an isomorphism?
0
votes
0answers
55 views

$\mathrm{SL}_2$ acts naturally on binary forms

Problem: Let $F^{(n)}$ be the vector space of binary forms of degree $n$ in two variables with coefficients in $\mathbb{C}$. Show that $\mathrm{SL}_2$ acts on $F^{(n)}$ and ...
1
vote
4answers
46 views

Show that $G$ is a group if the cancellation law holds when identity element is not sure to be in $G$

Having already read through show-that-g-is-a-group-if-g-is-finite-the-operation-is-associative-and-cancel, however in Herstein's Abstract Algebra, I was required to prove it when we're not sure if ...
5
votes
1answer
145 views

The field of algebraic numbers in $\mathbb Q (a_1,\ldots, a_l)$ is finite over $\mathbb Q$

In the book 'Algebra IV: Infinite Groups, Linear Groups' by Kostrikin and Shafarevich, there is a sketch of a proof (on page 84) of a theorem by Schur. I'm struggling to understand the line: Since ...
1
vote
2answers
39 views

Find $b \in G$ for each $a \in G$, such that $b^m = a$, where $m$ is coprime to the order of group G

Let $(G, \cdot)$ be a finite group (with identity element $1 \in G$) and $m \in \mathbb{N}$ be coprime to $|G|$. Proofe that for all $a \in G$ there exists exactly one $b \in G$, such that $b^m = a$. ...
2
votes
2answers
43 views

For a shift matrix $A$, prove that $A^n=0$ but $A^{n-1} \neq 0$.

Let $A\in F_n$ be the matrix $\begin{pmatrix} 0&1&0&0&\cdots&0 \\ 0&0&1&0&\cdots&0 \\ \vdots\\ 0&0&0&0&\cdots&0 \end{pmatrix}$, whose ...
0
votes
1answer
35 views

Nilpotent group and center

Let $G$ a nilpotent group. Show that (i) If $N$ is normal to $G$ and $N \neq 1$, then $N \cap Z(G) \neq 1$ (ii) If $N$ is normal to $G$ with $|N|=p$, where $p$ is a prime number, then $N \le Z(G)$. ...
1
vote
1answer
40 views

Prove that the Polynomial Ring $F[t]$ Is an Integral Domain

Let $F$ be a field and $F[t]$ be the ring of polynomials. Take $p(t)$, $q(t) \in F[t]$, and write $p(t) = \sum_{i=1}^{n} {a_i}{t^i}$, $q(t) = \sum_{k=1}^{n} {b_k}{t^k}$, where $n$ denotes the greater ...
1
vote
0answers
38 views

Classifying quadratic extensions of $\mathbb{Q}$

I'm studying Artin's Algebra, and the question says to "Classify quadratic extensions of $\mathbb Q$." What would that look like? A quadratic extension of $\mathbb Q$ is just, for $d$ square-free, ...
0
votes
1answer
26 views

Show that the equivalence classes of $\sim$ are left cosets of $H$ in $G$.

Let $H \leq G$ and define a relation on $G$ by $x \sim y$ if $y^{-1}x \in H$. Show that $\sim$ is an equivalence relation on $G$ and then show that the equivalence classes of $\sim$ are left cosets ...
0
votes
1answer
35 views

Show that $T(X) =\{f\in A(S)|f(X)\subset X\}$ is a subgroup of $A(S)$ if X is finite.

If $S$ is a nonempty set and $X\subset S$, Show that $T(X)=\{f\in A(S)|f(X)\subset X\}$ is a subgroup of $A(S)$ if X is finite. Note: $A(S)$ is called "symmetric group". It's actually a collection of ...
2
votes
3answers
61 views

The meaning of $\bigcap_{x\in G} x^{-1}Hx$ and the proof for the fact that $N$ is a subgroup of $G$ such that $y^{-1}Ny=N$ for every $y\in G$

If $H$ is a subgroup of $G$, let $N=\bigcap_{x\in G}x^{-1}Hx$. Prove that $N$ is a subgroup of $G$ such that $y^{-1}Ny=N$ for every $y\in G$. What does $N=\bigcap_{x\in G}x^{-1}Hx$ mean? I'm confused ...
0
votes
2answers
53 views

Prove that $ AB$ is a subgroup of order $mn$ if $m$ and $n$ are relatively prime.

If $A$ and $B$ are finite subgroups, of orders $m$ and $n$, respectively, of the abelian group $G$, Prove that $AB$ is a subgroup of order $mn$ if $m$ and $n$ are relatively prime. Definition: ...
1
vote
1answer
25 views

Let E be defined over Fq and let n ≥ 1. Show that E(Fq)[n] and E(Fq)/nE(Fq) have the same order.

Let E be an Elliptic Curve defined over $F_q$ and let n ≥ 1. Show that $E(F_q)[n]$ and $E(F_q)$/$nE(F_q)$ have the same order. I feel like this is obvious. The n-th torsion group $E(F_q)[n]$ ...
3
votes
2answers
30 views

Prove that $\lambda_a$ is a permutation of a group $G$ for a fixed element $a \in G$.

Hi I am working on following hw problem and I want to make sure that I am doing this correctly? I think I am going about this in the right way but I still need some reassurance. Let $G$ be a group ...
0
votes
5answers
108 views

Is this a true theorem? [closed]

I'm trying to prove the existence of the following theorem: If $n,p \in \mathbb{N}$, then $(p+1)^n = 1 \mod p$ Is this theorem true? I think it is, but I don't know how to prove it! Thanks!
1
vote
2answers
38 views

How can you find all subgroups of a certain group?

For example, how can how find all subgroups of $S_3$? How can you ensure that your answer includes all of them? Note: I want to know if there is a universal method to find all subgroups for a single ...
2
votes
2answers
43 views

$F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of F have a square root.

$F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of $F$ have a square root. (Hint: For some integer $i$, ($1\leq i \leq s-1$), $i$ is odd if and only if ...
1
vote
2answers
39 views

Given a finite abelian group $G$ with $g \in G$, then for any divisor $d$ of $|g|$ there is an element of $G$ with order $d$.

From an homework question that comes as an introduction to abelian groups. Regarding my efforts to solve the question, I have been trying to utilize the fundamental theorem of finite abelian groups, ...
0
votes
1answer
45 views

Set of elements of order 2 in a group.

For a group G, define $G_2$={ $g\in G$:$|g|=2$ }. Prove that if $G_2$ is finite, then $|G_2|$ is odd. Can you help me please?
3
votes
1answer
48 views

Simpler method to show $k(x)$ is NOT a primitive irreducible polynomial?

Let $k(x) = x^2+2x+2$ be under $\mathbb Z_{11}[x]$. Determine if $k(x)$ is irreducible, and if so, determine if it is primitive. Ok, I showed that $k(x)$ is irreducible since it it a quadratic and ...
0
votes
1answer
33 views

With B integral over subring A, homomorphism from A to algebraically closed field F can be extended to B.

Here's the problem I am working on: Let A be a subring of B such that B is integral over A, and let $f: A \rightarrow F$ be a homomorphism of A into an algebraically closed field F. Show that f ...
-1
votes
0answers
35 views

Subgroups with 300 elements [duplicate]

Show that every group which order 300 have a normal subgroup. I try show this for contradiction with Sylow's Theorem but I don't know what I have to do before that...
0
votes
1answer
28 views

Radical of an ideal - prove every prime ideal that contains $I$ also contains $\sqrt{I}$

Let $R$ be a commutative ring with a unit and $I$ an ideal. Please prove that every prime ideal that contains $I$ also contains $\sqrt{I}$. I easily conclud that $I \subseteq \sqrt I$ but I ...
0
votes
3answers
88 views

showing $Q[\sqrt 2] = Q(\sqrt 2)$

The question came in my exam. $Q[\sqrt 2] = \{ a + b \sqrt2 \;| a,b \in Q\}$ and $Q(\sqrt 2)$ is minimal subfield of it's extension containing $Q$ and $\sqrt 2$. (In my book) It calls $F(a)$ ...