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.

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 […]

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.

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

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 […]

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!

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.

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 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 […]

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 […]

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