Questions tagged [finite-groups]

Question 1

I just constructed a group of order 5 that is not abelian. Clearly, something is theoretically wrong, but I can't figure out what.

Question 2

I'm currently studying different aspects of the braid group, and I've come across various different definitions. In particular, I don't understand what is the connection between the pure braid group (...

Question 3

(NB: I'm aware that there's plenty of proofs for the Theorem: here I'm rather interested in whether the guessed argument can be brought to an end.) Trying to generalize the argument in the Example, I'...

Question 4

Here $p>2$ is a prime, the group $UT(3,p)$ is the group of $3\times3$ upper unitriangular matrices with coefficients in $\mathbb{F}_p$: $$\begin{bmatrix} 1&x&y\\ 0&1&z\\ 0&0&...

Question 5

This question is distantly related to the following MathStack post: How "big" can the center of a finite perfect group be? The above post and its answers comment on the size of the center of ...

Question 6

For $i=1,2$, let $B_i\in\mathrm{SL}_n(\mathbb{Z})$ be two symmetric positive definite matrices. We define their automorphism groups as $$\mathrm{Aut}(B_i)=\{g\in\mathrm{GL}_n(\mathbb{Z})\mid\ gB_ig^{\...

Question 7

Let $G$ = $NK$ be a semidirect product of N and K. Let $\sigma$ be the mapping from $G$ to $Aut(N)$ given by $\sigma(g)$ = $gng^{-1}$. I'm trying to use GAP to find $\sigma$ and $\sigma(K)$. Is ...

Question 8

Let G be a group acting on a set A. Let $[x]$ denote the orbit of any $x\in A$. Also let $G_x$ denote the stabilizer of $x$. From the orbit-stabilizer theorem, the orbit of any $x\in A$ has the same ...

Question 9

Let $G$ be a finite group. Define $\pi(G)$ to be the number of distinct prime factors of $|G|$. It is known that any finite group $G$ with $\pi(G)\leq 2$ is solvable. Also there exists many non-...

Question 10

Let $G$ a finite group, $p$ a prime number, $P$ a non trivial $p$-Sylow group of $G$ (i.e., $\vert P \vert =p^n$ with $n \ge 1$ for $\vert G \vert =p^nm$ with $(p,m)=1$) and $Q \leq G$ any $p$-group. ...

Question 11

This was a question posed at the end of a problem sheet in a group theory class I am taking: Problem Let $G$ be any group, let $g$ be any element of $G$ of finite order, and let $p$ be any prime. Show ...

Question 12

Let $A, B$ be finite groups. Suppose $A \triangleleft B$ with $B$ transitively acting upon $A \setminus \{1\}$ by conjugation. (This implies $A \cong (\mathbb{F}_p^n, +)$.) Must there exist $C$ with $...

Question 13

Let $G$ a finite group, $p$ a prime number, $P$ a non trivial $p$-Sylow group of $G$ (i.e., $\vert P \vert =p^n$ with $n \ge 1$ for $\vert G \vert =p^nm$ with $(p,m)=1$) and $Q \leq G$ any $p$-group. ...

Question 14

https://web.mat.bham.ac.uk/D.A.Craven/docs/lectures/finitegroups2010.pdf As shown in page 48 of Theorem 3.30 in above link. In the study of generalized Fitting subgroup of $G$. Let $E(G)$ be the layer ...

Question 15

It is known that for any finite simple group $G$ there exist two elements $a,b\in G$ such that $a$ and $b$ are conjugates in $G$ and $\langle a, b \rangle=G$. My question: Is it true for any finite ...

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Questions tagged [finite-groups]

Non-abelian group of order five? [closed]

Infinitesimal vs pure braid relations

Cyclicity of the multiplicative group of the integers modulo a prime

Automorphism group of $UT(3,p)$

Center of a finite perfect group

Automorphisms of integral quadratic forms

In a semidirect product $NK$, can GAP find the subset of $Aut(N)$ induced by the elements of $K$?

The consequences of the orbit-stabilizer theorem

Minimal finite non-solvable groups whose order has exactly three distinct prime factors.

A Consequence from Sylow Theorems on Conjugacy of all $p$-Sylow groups

Decomposition of finite-order element into commuting components of coprime order

Question about transitive conjugation actions

Consequence from Sylow Theorems on Conjugacy of all $p$-Sylow groups

Conditions on $F(C)=F(G)$ for Bender's theorem

Generating a finite non-solvable group with an element and its conjugate

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