Articles of abstract algebra

A group that has a finite number of subgroups is finite

This question already has an answer here: Finite number of subgroups $\Rightarrow$ finite group 2 answers

Kernel of action of G on set of cosets of H in G is contained in H

Let $G$ be a group and $H$ a subgroup of G. Also, let $X$ be the set of left cosets, $xH$, of H in G. Define an action of $G$ on $X$ by $g \cdot xH = gxH$ for $g,x \in G$. I have shown that the kernel, $K$, where $K=\bigcap_{xH \in X}xHx^{-1}$ and $Stab_{G}(xH) […]

centralizer of transvection

let $T=T_{a,u}(v)$ be transvection, I want to find centralizer of $T_{a,u}(v)$ in $GL(V)$. What is $C _{GL(V)}(T)$? $T_{a,u}(v)=v+u(v)a$ where $a$ is vector in vector space $V$ on field $F$ and $u$ is linear functional on $V$.It is not hard to see $T_{a,u}(v)\in SL(V)$ I have not any idea how to deal with it.

Find the number of integers $r$ such that the polynomial $x^{r}-a$ has a linear factor over $\mathbb{F}_{p^{n}}$

If we have a finite field $\mathbb{F}_{p^{n}}$, how does one determine the number of integers $r$ in $\{0,1, \ldots, p^{n}-2 \}$ for which the equation: $x^{r}=a$ has a solution for every $a \in \mathbb{F}_{p^{n}}$. The problem also mentions that $p^{n}-1=q_{1}^{a_{1}}\ldots{q_{n}^{a_{n}}}$ for distince primes $q_{i}$. My attempt to understand the problem: I tried to understand the […]

if $Q$ and $P$ are distinct $p$-Sylow subgroups then $Q\not\subseteq N_G(P)$.

I have been told to use the following to prove another claim, but I would like to prove this anyway for myself. However I can’t tell why it’s true. I think it’s true, but can’t see why! Here it is: If $P$ and $Q$ are distinct $p$-Sylow subgroups in a group $G$ then $Q\not\subseteq N_G(P)$. […]

Let R be a commutative ring, and let P be a prime ideal of R. Suppose that P has no nontrivial zero divisors in it. Show that R is an integral domain.

Let R be a commutative ring, and let P be a prime ideal of R. Suppose that P has no nontrivial zero divisors in it. Show that R is an integral domain. My proof: Take $r,s,a \in R$ with $ar = as$, and $p \in P$. Then \begin{align*} par &= pas\\ a p r &= […]

Determine the degree of the extension $\mathbb{Q}(\sqrt{3 + 2\sqrt{2}})$.

The following is from a set of exercises and solutions. Determine the degree of the extension $\mathbb{Q}(\sqrt{3 + 2\sqrt{2}})$ over $\mathbb Q$. The solution says that the degree is $2$ since $\mathbb{Q}(\sqrt{2}) = \mathbb{Q}(\sqrt{3 + 2\sqrt{2}})$. I understand that the LHS is a subset of the RHS since $$ \sqrt{2} = \frac{(\sqrt{3 + 2\sqrt{2}})^2 – […]

Cardinality of a $\mathbb Q$ basis for $\mathbb C$, assuming the continuum hypothesis

Prove that $\operatorname{tr.deg}(\mathbb{C/Q})=\mathfrak{c}$ (where $\mathfrak{c}$ is the cardinality of $\mathbb{R}$) using the continuum hypothesis. $Proposition$ If $E/F$ is an algebraic extension then $|E|=\aleph_0 |F|$ Proof: Let $S$ be a trancendental basis of the extension $\mathbb{C/Q}$.Then $|S| \leqslant \mathfrak{c}$.Suppose that $|S|< \mathfrak{c}$. Then, by the continuum hypothesis, $S$ is finite or countable. Let $S=\{s_1,s_2, \ldots \}$ […]

If $\operatorname{rk}(N)=1$, and $M/N$ is torsion, why is $\operatorname{rk}(M)=1$?

Suppose $M$ is a finitely generated torsion-free module over a PID. If $N\leq M$ is free of rank $1$, and $M/N$ is torsion, how do we conclude $M$ is free of rank $1$? My scattered thoughts: Since $M$ is f.g., $M/N$ is as well, and since it is torsion, the structure theorem tells me it […]

What does it mean to be an irreducible polynomial over a field? (need clarification on the definition)

My understanding is to find out if a polynomial is reducible, we can use substitution. For example, to see if $x^2+1$ is reducible over $F_2$, we can substitute $0$ and $1$ for $x$, and see that neither of them result in $0$ mod $2$. But we know $x$ is irreducible because it’s of degree $1$ […]