Intereting Posts

Can every group be represented by a group of matrices?
The ring $\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$
What is the difference between intrinsic and extrinsic curvature?
Prove that the additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic.
Why does $k/(XY)$ have two minimal primes?
If a derivative of a continuous function has a limit, must it agree with that limit?
Taking powers of a triangular matrix?
Prove that $Q_8 \not < \text{GL}_2(\mathbb{R})$
Cardinality of a basis of an infinite-dimensional vector space
Must subgroups sharing a common element be nested in each other?
Evaluate $ \int_{0}^{\frac{\pi}2}\frac1{(1+x^2)(1+\tan x)}\:\mathrm dx$
Consider the function, f and its second derivative:
Another pigeonhole principle question
Improper Integral $\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx$
Finding roots of 5 th degree taylor expansion of $e^x$

If $G$ is a non abelian finite simple group, we know that $4$ divides $|G|$.

More precisely there are infinitely many finite simple groups $G$ such that $v_2(|G|)=2$, just consider $\mathrm{PSL}_2(\mathbb F_p)$ with $p\equiv 3 \pmod 8$, $p$ prime and $p>3$.

So my question tries to generalize the above statement:

- How to define addition through multiplication?
- Detecting that a fraction is a repeating decimal
- Is the difference of two irrationals which are each contained under a single square root irrational?
- Comparing powers without logarithms
- Sum of the reciprocals of divisors of a perfect number is $2$?
- What combinatorial quantity the tetration of two natural numbers represents?

Do we know the primes $p$ and positive integers $n$ such that there

exists (or doesn’t exist) infinitely many finite simple groups $G$ such

that $v_p(|G|)=n$?

- Non-standard models of arithmetic for Dummies
- Union of the conjugates of a proper subgroup
- The use of conjugacy class and centralizer?
- How addition and multiplication works
- How to find presentation of a group using GAP?
- Why is $\frac{987654321}{123456789} = 8.0000000729?!$
- Irreducible representations (over $\mathbb{C}$) of dihedral groups
- Finite abelian $p$-group with only one subgroup size $p$ is cyclic
- Divisor of a finite group
- Is $\sqrt{x^2}=|x|$ or $=x$? Isn't $(x^2)^\frac12=x?$

I think you don’t need any new tricks, other than a Suzuki group. The short answer is that if $p$ is odd or $n\geq 2$, then there are infinitely many simple groups (of twisted Lie rank 1) whose order has $p$-adic valuation exactly $n$.

$\newcommand{\PSL}{\operatorname{PSL}}\newcommand{\Sz}{\operatorname{Sz}}$

For $p=2$ and $n\geq 2$ choose $q \equiv 2^n \pm 1 \mod 2^{n+2}$ by Dirichlet.

$$q^2-1 \equiv (2^n\pm1)^2 -1 \equiv 2^{2n}\pm2^{n+1} +1-1 \equiv 2^{n+1} \mod 2^{n+2},$$ so $v_2(q^2-1) = n+1$ and $v_2(|\PSL(2,q)|) = n$.

If $p$ is odd and $n \geq 1$, then choose $q\equiv p^n \pm 1 \mod p^{n+1}$, then $$q^2-1 \equiv (p^n \pm 1)^2-1 \equiv p^{2n} \pm 2\cdot p^{n} +1-1 \equiv \pm 2 p^n \mod p^{n+1},$$ and $v_p(\gcd(q-1,2)) = 0$, so $v_p( |\PSL(2,q)| ) = n$.

If $p > 3$ and $n=0$, then choose $q \not\equiv \pm1 \mod p$, then

$$q^2-1 \not\equiv 0 \mod p,$$

and $v_p(|\PSL(2,q)|) = n = 0$.

If $p=3$ and $n=0$, then choose $q=2^{2n+1}$ and then $v_3(|\Sz(q)|) = 0$, where $\Sz(q)$ is the Suzuki simple group defined as a subgroup of the group of Lie type B2 over the field of $q$ elements.

If $p=2$ and $n \leq 1$, then no simple group has order with 2-adic valuation $n$ by Feit-Thompson and Cayley (for $n=0$ and $n=1$ respectively).

- Solving functional equation $f(x)f(y) = f(x+y)$
- Intuition behind the definition of linear transformation
- 1-separated sequences of unit vectors in Banach spaces
- Can we create a measure for infinitely countable sets with respect to eachother?
- Fundamental Theorem of Calculus problem
- How discontinuous can the limit function be?
- Proof about fibonacci numbers by induction
- Why is $f(x) = x\phi(x)$ one-to-one?
- finding the combinatorial sum
- Integrate $\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$
- Markov Chain – method of generating function -to find nth power
- Question on primitive roots of unity
- How to show that $\gcd(n! + 1, (n + 1)! + 1) \mid n$?
- How to solve $a_n = 2a_{n-1} + 1, a_0 = 0, a_1 = 1$?
- Joint probability distribution (over unit circle)