Articles of group theory

About the converse of Maschke's theorem

The Maschke’s theorem say that\ Let $G$ be a finite group and $F$ a field whose characteristic does not divide $\mid G \mid$. Then every $FG$-module is completely reducible (I’m using the notation of Isaac’s in the book Character Theory of finite groups). The converse of this theorem is true, but I can not prove […]

A question on Groups and its center

If $G$ be a group of order $8$ and $o(x)=4$ then how to prove that $x^2 \in Z(G)$ ? I can only figure out that $x^2=x^{-2}$ ; Please help

Is there a simple way to distinguish between group homomorphisms?

More precisely, I am given a function $f:G\to H$ with the promise that it is a homomorphism. Is there an easy way to determine which homomorphism it is without looking through all of its values? For example, when $G=\mathbb{Z}_2^n$ and $H=\mathbb{Z}_2$, a homomorphism $f$ is entirely characterized by an element of $\mathbb{Z}_2^n$, $s\in \{0,1\}^n$, such […]

Subgroups of $S_{n}$

There’s the theorem that states that every finite group $G$ of order $n$ is isomorphic to a subgroup of $S_{n}$. My question is, “How do we find that subgroup?”

Group Actions: Orbit Space

Given a group action $G\curvearrowright X$. Consider the orbit space: $\pi:X\to X/G$ Do continuous group actions correspond to open projections, i.e.: $$l_g\in\mathcal{C}(X)\quad(g\in G)\iff\pi(U)\in\mathcal{T}_{X/G}\quad(U\in\mathcal{T}_X)$$ (Note that this is a slightly different more appropriate version of continuous group actions.) Certainly, continuous group actions give rise to open projections since: $$\pi^{-1}(\pi(U))=\bigcup_{u\in U}Gu=GU=\bigcup_{g\in G}gU=\bigcup_{g\in G}l_g(U)\in\mathcal{T}$$ Surely, the converse may […]

Do we conclude in that way that it is a $p$-Sylow subgroup?

I am looking at the following exercise: If $G$ is finite and $f:G\rightarrow H$ is a group epimorphism, show that if $P\in \text{Syl}_p(G)$ then $f(P)\in \text{Syl}_p(H)$. $$$$ I have done the following: Since $P\in \text{Syl}_p(G)$, we have that $P$ is a subgroup of $G$. Do we have from the correspondence theorem that $f(P)$ is a […]

Intuitive explanation for the connection between Lie Groups and projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$

In this post John Baez states that the classical simple Lie groups “arise naturally as symmetry groups of projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$”. Is there some intuitive explanation for this connection? I understand that $SO(n)$ is the set of all rotations about the origin of $n$-dimensional Euclidean space $\mathbb{R}^n$ I understand that the […]

Hidden subgroup problem for $\mathbb{Z} mod 2$

The definition of the Hidden Subgroup Problem (HSP) is as follows (according to a lecture series by Pranab Sen), Let $G$ be a group, $S$ a set and $f : G \to S$ a function. We are given an oracle for a reversible version of $f$ , $F(f) : |x\rangle |s\rangle \mapsto |x\rangle|s \oplus f […]

No Simple Group of Order 144

I see the proof here:, but I can’t follow it, so could someone please explain to me how this proof works? Or maybe offer an alternative proof?

Line graph of Cayley graph of $\mathbb{Z}_2^3$ is $A_4$

Consider the group $G=\mathbb{Z}_2^3$ with generators $S=\{e_1,e_2,e_3\}$ with $e_1=(1,0,0),e_2=(0,1,0),e_3=(0,0,1)$. The Cayley graph $\text{Cay}(G,S)$ is the 3D hypercube graph. It’s line graph $\Gamma$ is the $4$-regular Cuboctahedron graph. This graph is the Cayley graph of $A_4$ with generators $g=(1 2 3), h=(234)$ (as noted here). I want to understand the fact the that $\Gamma$ is the […]