Intereting Posts

Is the Bourbaki treatment of Set Theory outdated?
Elementary set theory homework proofs
Sequence problem involving inequalities
Size of a union of two sets
Applications of conformal mapping
existence of a special function
My proof of “the set of diagonalizable matrices is Zariski-dense in $M_n(\mathbb F)$”.
Example of a group where $o(a)$ and $o(b)$ are finite but $o(ab)$ is infinite
Proving The Extension Lemma For Vector Fields On Submanifolds
prove inequlity about cardinality power sets
Can mathematical definitions of the form “P if Q” be interpreted as “P if and only if Q”?
Riemann Integral Upper vs Lower Estimate. Inf vs sup?
Proving Theorems
Summation of series $\sum_{n=1}^\infty \frac{n^a}{b^n}$?
What's the difference between early transcendentals and late transcendentals?

How to prove that $Aut(PSL(2,7))=PGL(2,7)$? Is this result extendible to $PSL(n,q)$ where $q=p^n$?

- Intuition behind quotient groups?
- If $p$ and $q = 2p + 1$ are both odd primes, show that $-4$ and $2(-1)^{(1/2)(p-1)}$ are both primitive roots modulo $q$.
- homomorphism $f: \mathbb{C}^* \rightarrow \mathbb{R}^*$ with multiplicative groups, prove that kernel of $f$ is infinite.
- What is the center of a semidirect product?
- If $x^m=e$ has at most $m$ solutions for any $m\in \mathbb{N}$, then $G$ is cyclic
- Is there a name for a group having a normal subgroup for every divisor of the order?
- Is $SO_n({\mathbb R})$ a divisible group?
- Could the concept of “finite free groups” be possible?
- Every abelian group of finite exponent is isomorphic to a direct sum of finite cyclic groups?
- Finitely Presented is Preserved by Extension

OK, here is a quick direct argument to show that Aut(PSL$(2,7)$) = PGL$(2,7)$. Let $\alpha \in {\rm Aut}({\rm PSL}(2,7)$. Now PSL$(2,7)$ acts 2-transitively by conjugation on its 8 Sylow $7-$subgroups. Since these must also be permuted under the action of $\alpha$, $\alpha$ is induced by conjugation by an element $a \in S_8$. By multiplying $a$ by an inner automorphism, we may assume that it fixes a specific Sylow $7-$subgroup $S$ of PSL$(2,7)$. Since the full automorphism group of $S$ (i.e. the cyclic group of order 6) is induced on $S$ within PGL$(2,7)$, by

multiplying $a$ be an element of PGL$(2,7)$, we may assume that $a$ centralizes $S$. Then, by multiplying $a$ by an element of $S$, we can assume that $a$ fixes some other point as an element of $S_8$, but then the fact that it centralizes $S$ forces $a=1$.

The same argument works for ${\rm Aut}({\rm PSL}(2,p)$ for any odd prime $p$. For ${\rm PSL}(n,p^k)$ with $k>1$ there are also field automorphisms, and for $n \ge 3$, there is also the graph automorphism (induced by inverse-transpose on ${\rm SL}(n,p^k)$), but you would probably need to learn some more general theory of classical groups or groups of Lie type to understand that.

- A set is infinite iff it is equivalent to a proper subset of itself
- How to study math to really understand it and have a healthy lifestyle with free time?
- the composition of $g \circ f$ is convex on $\Omega$ from Shapley's definition of convexity
- Convergence in distribution for $\frac{Y}{\sqrt{\lambda}}$
- Finding the no. of possible matrices given the order and limited no. of entries
- Decomposite a vector field into two parts
- second derivative at point where there is no first derivative
- Find general solution for the equation $1 + 2 + \cdots + (n − 1) = (n + 1) + (n + 2) + \cdots + (n + r) $
- Sequence of partial sums of e in Q is a Cauchy sequence.
- Proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ (and now $\pi^{\left(\pi^{\pi^\pi}\right)}$ is a noninteger
- What are spinors mathematically?
- $\sum_{k=-\infty}^\infty \frac{1}{(k+\alpha)^2} = \frac{\pi^2}{\sin^2\pi \alpha}$
- Intuitively and Mathematically Understanding the Order of Actions in Permutation GP vs in Dihereal GP
- How to obtain equation of line of the form $ax + by + c = 0$?
- Simplify $\int_0^\infty \frac{\text{d}{x}}{e^x+x^n}$