Articles of abstract algebra

What is the best way to self-study GAP?

Background: This year I’ll do another Group Theory course ( Open University M336 ). In the past I have used Mathematica’s AbstractAlgebra package but (although visually appealing ) this is no longer sufficient (i.e. listing subgroups of $S_4$ takes ages). So, I want to learn more about GAP. I worked through beginner tutorials that I […]

Projective and injective modules; direct sums and products

I need two counterexamples. First, a direct sum of $R$-modules is projective iff each one is projective. But I need an example to show that, “an arbitrary direct product of projective modules need not be a projective module.” If I let $R= \mathbb Z$ then $\mathbb Z$ is a projective $R$-module, but the direct product […]

Some irreducible character separates elements in different conjugacy classes

Let $x$ and $y$ be elements that are not conjugate in $G$. Then there is some irreducible character $\chi$ such that $\chi(x) \not = \chi(y)$. Clearly the “irreducible” part isn’t important, since any character can be written as the sum of irreducible characters, but I’m having trouble going beyond that. I’d appreciate a good hint […]

Question about Axler's proof that every linear operator has an eigenvalue

I am puzzled by Sheldon Axler’s proof that every linear operator on a finite dimensional complex vector space has an eigenvalue (theorem 5.10 in “Linear Algebra Done Right”). In particular, it’s his maneuver in the last set of displayed equations where he substitutes the linear operator $T$ for the complex variable $z$. See below. What […]

Ring homomorphisms $\mathbb{R} \to \mathbb{R}$.

I got this question in a homework: Determine all ring homomorphisms from $\mathbb{R} \to \mathbb{R}$. Also prove that the only ring automorphism of $\mathbb{R}$ is the identity. I know that $\mathbb{R}$ is a field, so the only ideals are $\mathbb{R}$ and $\{0\}$. Therefore the homomorphisms must be the identity and the function $f(x)=0$ where $x […]

Generating the symmetric group $S_n$

I know that $\sigma =(1 2 \ldots n)$ and $\tau =(1 2)$ together should generate the symmetric group by virtue of conjugation, i.e. $(\sigma)^k \circ \tau \circ (\sigma^{-1})^k = (k+1, k+2)$; we know that the set of adjacent transpositions generates $S_n$, so we’re done. However– and I realize that this question is incredibly dumb– when […]

Any group of order $n$ satisfying $\gcd (n, \varphi(n)) =1$ is cyclic

Sorry for the last mistaken problem I just posted. Now I know that only having the order being odd square free is not enough for a group to be cyclic. Here’s the complete problem which the main goal is to show that any group of order $n$ satisfies $\gcd (n,\varphi(n))$=1 is cyclic. It asks me […]

Does a non-abelian semigroup without identity exist?

I was introduced to semigroups today and had a question. So all the examples of semigroups I was given were either monoids or groups. So I was curious, does there exist a semi-group which is not abelian and does not contain identity? I tried to construct an example, but every example I tried to construct […]

Is this quotient Ring Isomorphic to the Complex Numbers

So the question goes: Let $$ A=\mathbb{R}[x]/\langle x^2-x+1\rangle . $$ Is A isomorphic to $ \mathbb{C} $ ? The earlier parts of the question asked for me to a) find the reciprocal in A of $x+1+I$ and b) find $p(x)+I\in A$ such that $(p(x)+I)^2=-1+I$. I found the answers to both of these parts, but I […]

When $\Bbb Z_n$ is a domain. Counterexample to $ab \equiv 0 \Rightarrow a\equiv 0$ or $b\equiv 0\pmod n$

Suppose $$ab \equiv 0 \mod n$$ and that $a$ and $b$ are positive integers both less than $ n$. Does it follow that either $a | n$ or $b | n$? If it does follow, give a proof. If it doesn’t, then give an example. For this question is this a suitable answer: No it […]