### abstract algebra - Using GAP to Identify a Group

How do you use GAP to identify the name of a group from its multiplication table? I know that you can define a group from a set of generators, and then look for the group in the set of internal tablesgap> g := Group([ (1,2), (1,2,3,4,5) ]); Group([ (1,2), (1,2,3,4,5) ])gap> IdGroup(g); [ 120, 34 ]But how do find out the name of group [120, 34]?...Read more

### abstract algebra - Assume that $\forall a\in R, a^2+a\in \operatorname{cent}R$. Need to show that $R$ is a commutative ring.

Let $R$ be a ring. Assume that $\forall a\in R, a^2+a\in \operatorname{cent}R$. I need to show that $R$ is a commutative ring. The author gives a hint; that is to show that $\forall a,b\in R, ab+ba\in \operatorname{cent}R$, and I did show that it happens. But I don't know how to proceed. Could someone please give me a second hint? Thank you....Read more

### abstract algebra - Memorising lots of maths theorems/lemmas

In a few weeks I'll have my summer exams in a Senior Freshman Mathematics course. Two of my modules have a huge number of theorems, lemmas and definitions - over 200 definitions and 250 proofs by my last count - and I'm struggling to find a way to memorise them all. I know the key to memorisation is to understand the topic (which I do) however I still need to be able to perfectly recall the exact proof/definition in question. (It doesn't help that a lot of the material is similar and can be confused with a different proof/definition.) So my que...Read more

### learning - How to study abstract algebra

I am taking Abstract Algebra course at the university. We are doing chapters 1-20 from Gallian's abstract algebra text book.I am just doing assigned homework everyweek ( About 5 questions from each chapter). Although I am getting an A in all the assignments and midterms, but I am really worried that my understanding might be shallow or just enough to do the homework that I am going to promptly forget when the course is over. My question is how do I know that I am gaining knowledge that would stay with me and not just studying enough to do the a...Read more

### abstract algebra - Intuition on the Orbit-Stabilizer Theorem

The Orbit-Stabilizer says that, given a group $G$ which acts on a set $X$, then there exists a bijection between the orbit of an element $x\in X$ and the set of left cosets of the stabilizer group of $x$ in $G$. In other words, that the cardinality of the orbit of an element $x\in X$ is equal to the index of its stabilizer subgroup in $G$. I've seen two different texts present this, both of which explicitly say that this captures a very intuitive idea. I'm sorry if it's obvious, but I don't see the intuition behind this. I've asked a few questi...Read more

### abstract algebra - Group of sphere transformations, impressing friends

Ok, so here's the story: I am reading a book on algebra and, via some exercises, discovered that in any group $G$, the order of $x \cdot y$, written $o(x \cdot y)$, equals $o(y \cdot x)$. Now, this is trivial in an abelian group, but I was looking for examples of a non-abelian group (simply because the result was interesting) to see this happen.Of course, I knew $GL(2, \mathbb{R})$ and the permutation groups. However, literally by chance (I had a ball in my hand), I realized that $m(90)$ degree rotations of a sphere - $m \in \mathbb{N}$ - are a...Read more