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

abstract algebra - Show that $\operatorname{Aut}(C_{14})$ is a cyclic group.

In the context of abstract algebra and groups automorphism, I need help to understand the solution to this problem: Show that $\operatorname{Aut}(C_{14})$ is a cyclic group. Here's the solution (I'm directly translating from Greek handwritten notes so I hope there are no mistakes):Since $C_{14}$ is cyclic with $14$ elements, $C_{14} \cong Z_{14}$. Hence, it suffice to show that $\operatorname{Aut}(\mathbb{Z}_{14})$ is cyclic. Also, $\mathbb{Z}_{14} = \{0, 1, 2, 3, \ldots 13\} = \langle 1 \rangle = \langle 3 \rangle = \langle 5 \rangle = \langl...Read more

abstract algebra - how can we define real number to form of a (first-order) structure of type $\mathcal L$?

We know that : Let $\mathbb{R}$ denote the set of all real numbers. Then: 1-) The set $\mathbb{R}$ is a field, meaning that addition and multiplication are defined and have the usual properties. 2-)The field $\mathbb{R}$ is ordered, meaning that there is a total order ≥ such that, for all real numbers $x, y$ and $z$: if $x ≥ y$ then $x + z ≥ y + z$; if $x ≥ 0$ and $y ≥ 0$ then $xy ≥ 0$. The order is Dedekind-complete; that is: every non-empty subset $S$ of $\mathbb{R}$ with an upper bound in $\mathbb{R}$ has a least upper bound...Read more

abstract algebra - Euclidean Algorithm for differential operators

While perusing through the article "Algorithms for solving linear ordinary differential equations" by Winfried Fakler (a pdf can be found through a google search), I noticed Faker mentioning on page 2 that From a mathematical point of view linear differential operators generate a left skew polynomial ring of derivation type. The elements of such a ring are called skew polynomials or Ore polynomials. For Ore polynomials the usual polynomial addition holds. Only the multiplication is different. It is declared as on extension of the rule $a\in k$...Read more

reference request - Resources for studying differential equations through algebra

I come from an applied math background and my interest lately have been in the application of PDE theory to machine learning problems. This has led me down a more "pure" route concerning solutions of differential equations when the coefficients come from more general algebraic structures (such as rings).A simple example comes in the one dimensional case.$$y' - \lambda y = 0$$Which has solutions of the form$$y = y_0 e^{\lambda x}$$Now if we look at the similar equation$$D_x \mathbf{Y} - \Lambda\mathbf{Y} = \mathbf{0}$$As long as $\Lambda$ is a c...Read more

soft question - The most common theorems taught in Abstract Algebra

I am self learning abstract algebra. I want to know which theorems are a must to understand. Now these are limits I have to deal with (please consider when answering): I have limited internet access Few mathematical books written in English are available. I can not afford to order books from abroad.  I just want to know what is the core knowledge (theorems, lemmas, etc) of any decent graduate level abstract algebra class....Read more

abstract algebra - Is there some sort of correspondence between groups and partitions of a set?

Every group action on a set $S$ partitions the set into orbits. Conversely, for every partition of $S$ is there a group action such that the set of orbits of the group action equals the partition?My attempt. Let $P = \{S_1,\dots, S_k\}$ be a partition of a finite set $S$ of order $n$. Then there are elements of $G = Perm(S)$, the set of permutations of $S$, that leave each subset fixed, i.e. $g\in G$ such that $g\cdot S_i = \{gs : s\in S_i\} = S_i, \forall i$. Think of the permutations that fix all of $S$ except permutes $S_i$ possibly in ...Read more