Articles of group theory

The value of the rational “Möbius”-like transformations at infinity

If $\mathbb F$ is some field, the group $PSL(2, \mathbb F)$ consists of the mappings $$ x \mapsto \frac{ax + b}{cx + d} $$ with $a,b,c,d \in \mathbb F$ and $ad – bc = 1$. These mappings are defined over the extended field $\mathbb F_{\infty} := \mathbb F \cup \{\infty\}$ (or the so called projective […]

Confused about this exercise question: if a set with a certain binary operation is a group

I tried to answer the following exercise: Let $S$ be a nonempty set with an associative operation that is left and right cancellative. Assume that for every $a$ in $S$ the set $\{a^n \mid n=1,2,3, \dots \} $ is finite. Must $S$ be a group? My thoughts: This is supposedly an old Putnam question so […]

Birational Equvalence of Twisted Edwards and Montgomery curves

I’m trying to understand the birational equivalence between Twisted Edwards and Montgomery curves and try to calculate some examples. In particular, as an example, I’m looking at the Ed25519 Twisted Edwards curve: $$ax^2 + y^2 = 1 + dx^2 y^2 \quad \text{where} \quad a = -1,\ \ d = \frac{-121665}{121666} \quad\text{over the finite field}\quad F_{2^{255}-19}$$ […]

$C_G(x)$ in a solvable group

Let $G=PQ$ be a solvable group with $P$ and $Q$, P- and q-sylow subgroup of $G$ respectively. Suppose both $P$ and $Q$ are not normal and $C_G(P)=Z(G)$ and $C_G(Q)>Z(G)$. Let $x\in C_G(Q)-Z(G)$. So $Q\leq C_G(x)$. Is it true that such $x$ is a $q$-element?

Embedding $G$ in a $Z(G)$ extension of $\operatorname{Aut}G$.

The present question follows up this one, in which I accidentally asked for less than I actually wanted. Given a group $G$, I would like to find an extension $\tilde G$ of its automorphism group $\operatorname{Aut}G$ by its center $Z(G)$, into which $G$ embeds in such a way that the restriction of the surjection $\tilde […]

0n group that have an non-trivial element fix with each automorphism

Let $G$ be a group and $Aut(G)$ is an automorphisms groups of $G$. We know that if $Aut(G)$ is nilpotent and $G$ is not cyclic of odd order, then $G$ has an non-trivial element such that fix by all automorphisms. Also it is clear that if $G$ has a characteristic subgroup of order 2, then […]

Is every function between finite sets a restriction of a morphism of finite abelian groups (up to bijection)?

Hopefully this is a quick question with an easy answer. Can every function between finite sets be written as the restriction of a group homomorphism of finite abelian groups, up to bijection of the domain and codomain? More explicitly, it it the case that for every $f : X \to Y$ with $X, Y$ finite […]

Size of the product of two subgroups

Let $(G, \ast)$ be a group and let $H\le G$ and $K\le G$ be subgroups of $G$. Prove that $|HK|$=$\frac{|H|\cdot|K|}{|H\cap K|}$. Intuitively this is quite obviously true, as otherwise the products of all elements in the intersection of $H$ and $K$ would be counted twice, but no idea how to prove it! Any advice appreciated.

Application of Sylow's theorem

Let $p>q$ be primes. $ (1): \exists $ non-abelian group of order $pq$ $\Longleftrightarrow$ $p \equiv 1 (mod \ q)$ $(2):$ Any $2$ non-abelian groups of order $pq$ are isomorphic to each other. Proof of claim $(1):$ Suppose $\exists$non-abelian $G$ of order $pq.$ Let $P$ be the $p-$sylow subgroup of $G$ and $N(P)$ be the […]

Isomorphism between a quotient group and the 2-torsion subgroup

Let $G$ be a finite abelian group, and let $2G$ denote the subgroup $\{ g * g : g \in G\}$. Let $G[2]$ be the 2-torsion subgroup of $G$. I want to show that $$ G/2G \cong G[2]. \qquad (1) $$ The closest I could get was to prove that $G/G[2] \cong 2G$ using the […]