Articles of abstract algebra

Is there a systematic way of finding the conjugacy class and/or centralizer of an element?

Is there a systematic way of finding the conjugacy class and centralizer of an element? Could the task be simplified if we are working with “special groups” such as $S_n$ or $A_n$? Are there any intuitive approaches? Thanks.

A finite Monoid $M$ is a group if and only if it has only one idempotent element

Suppose that $(M,*)$ is a finite Monoid. Prove that $M$ is a group if and only if there is only a single idempotent element in $M$, namely $e$. One direction is obvious, because if $M$ is a group then $x^2=x$ implies $x=e$, but the other direction has been challenging me for over an hour, so […]

Efficiently find the generators of a cyclic group

Is there any efficient method to find the generators of a cyclic group? Edit: The (cyclic)group here refers to a general multiplicative group of prime modulo. Is there any efficient algorithm to find the generators of the (cyclic)group.

Direct summand of a free module

Let $M$, $L$, $N$ be $A$-modules and $M=N\oplus L$. If $M$ and $N$ are free, is $L$ necessarily free?

Prove that in any GCD domain every irreducible element is prime

The proof of the following proposition is not completely clear to me. I get everything up until the bold part and I have a feeling some crucial steps are omitted, can anybody help clear this up? Let $R$ be an integral domain. If every two elements of $R$ have a greatest common divisor, then every […]

Bijection between ideals of $R/I$ and ideals containing $I$

I read that there is a one-one correspondence between the ideals of $R/I$ and the ideals containing $I$. ($R$ is a ring and $I$ is any ideal in $R$) Is this bijection obvious? It’s not to me. Can someone tell me what the bijection looks like explicitly? Many thanks for your help!

Galois group of algebraic closure of a finite field

Is there any element of finite order of the Galois group of algebraic closure of a finite field, and if there is how can I construct it ? Thanks.

Show that $a \star b=a \cdot b+a+b$ is binary operation for the group $\Bbb{ Q} – \{-1\}$

The group $\left(\Bbb{ Q} – \{-1\},\star\right)$ has as its underlying set the rational numbers different from $-1$ and the operation $\star$ is defined as $a \star b=a \cdot b+a+b$ where multiplication/addition are the usual operations with rational numbers.Show that this is a binary operation. What I did (Please help verify): identity element has to be […]

Describe all ring homomorphisms

Describe all ring homomorphisms of: a) $\mathbb{Z}$ into $\mathbb{Z}$ b) $\mathbb{Z}$ into $\mathbb{Z} \times \mathbb{Z}$ c) $\mathbb{Z} \times \mathbb{Z}$ into $\mathbb{Z}$ d) How many homomorphisms are there of $\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}$ into $\mathbb{Z}$ Note: These were past homework questions and my professor already gave out answers. I just need someone to help me […]

How to tell if some power of my integer matrix is the identity?

Given an $n\times n$-matrix $A$ with integer entries, I would like to decide whether there is some $m\in\mathbb N$ such that $A^m$ is the identity matrix. I can solve this by regarding $A$ as a complex matrix and computing its Jordan normal form; equivalently, I can compute the eigenvalues and check whether they are roots […]