Articles of abstract algebra

What's group $E(2)$ of isometries of $\mathbb{R}^2$?

I’m trying to prove normal subgroups of the group $E(2)$, but I haven’t been given, what the group $E(2)$ of isometries of $\mathbb{R}^2$ is like. What is it like?

How to show that if $k | n$, then $D_{2k} \leq D_{2n}$?

I can see why geometrically this is true. I have an idea where we generate a group with $<r^{n/k}, s>$, but I’m not sure how to complete this or whether this will really work.

Inductive vs projective limit of sequence of split surjections II

This question is a follow-up of this earlier question I asked. Let $$ A_1\twoheadrightarrow A_2\twoheadrightarrow A_3\twoheadrightarrow A_4\twoheadrightarrow \cdots $$ be an inductive sequence of abelian groups, the connecting homomorphisms of which are surjective and split, that is, we have embeddings $A_{n+1}\rightarrowtail A_n$ such that the diagram \begin{array}{ccccccccc} A_n & \twoheadrightarrow & A_{n+1}\\ \uparrow & & […]

$\forall m,n \geq 2$, $\exists$ non-cyclic group of order $n^{m}$

I need to prove that for any integers $m$, $n$ $\geq 2$, there exists a non-cyclic group of order $n^{m}$. I have a result that says that if $G$ is a finite group, and $G = H_{1}\times H_{2}$ then $G$ is cyclic iff both $H_{1}$ and $H_{2}$ are cyclic and $\gcd\left( |H_{1}|,|H_{2}|\right) = 1$. If […]

Rings of fractions for all possible sets of denominators

This is related to a question I asked here except in this case, I am not asking about the total (complete) ring of fractions. I am asking how to find all possible rings of fractions up to isomorphism for all possible denominators. Specifically, I am asking for the cases when the rings in question are […]

Where is the notion of anti-isomorphism useful

Let $(S,\cdot)$ and $(T,\circ)$ be semigroups (or some algebraic structure with an operation), then they are anti-isomorphic if there exists some $\varphi : S \to T$ such that $$ \varphi(xy) = \varphi(y) \circ \varphi(x). $$ Now for what is this notion useful? The notion of isomorphism is useful as this basically says that isomorphic structures […]

Group presentations – again

My question is about finding presentations for finite groups. It’s along similar lines to my earlier question — but is subtly different! The earlier question is here Group presentations: What's in the kernel of $\phi$? Let’s take a finite group, say $D_4$, and find a set of generators. Say we find two generators; denote them […]

Proof that the $(\mathcal P (\mathbb N),\triangle)$ is an abelian group?

This question already has an answer here: Does the Symmetric difference operator define a group on the powerset of a set? 3 answers

How does one compute a resultant by using Euclidean algorithm?

In an answer to the question Resultant of Two Univariate Polynomials, a PDF of course slides was linked which describes a modification of Euclid’s algorithm for computing univariate polynomial resultants (reference: – slide 53). The algorithm is as follows (with F, G the input polynomials, d_i the degree of F_i, and LC(p) the leading […]

Reference request: algebraic methods in geometry

I am a (soon to be) third year undergraduate who has just finished courses in linear and abstract algebra. While I enjoyed the study of algebraic structures in their own right, my favorite part of the courses were the applications of the algebraic machinery developed to geometric problems (i.e. the connection between Galois theory and […]