Intereting Posts

Divisibility of discriminants in number field extensions
Why do we find Gödel's Incompleteness Theorem surprising?
Natual density inside a subsequence
Discriminant for $x^n+bx+c$
Prove $n\binom{p}{n}=p\binom{p-1}{n-1}$
Confused about complex numbers
Proof that $2^{\mathbb N}$ and $\mathbb R$ have same cardinality
Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$
Why don't all odd functions integrate to $0$ from $-\infty$ to $\infty$?
How does one prove that a multivariate function is univariate?
Random walk $< 0$
Why is the derivative of a circle's area its perimeter (and similarly for spheres)?
Limit of the sequence $\lim_{n\rightarrow\infty}\sqrtn$
For each continuous $g:X\to $, $g(a_n)\to g(a)$, can we deduce $a_n\to a$?
Cannot calculate antiderivative using undefined coefficients

I want to show that there are no simple groups of order $p^{k}(p+1)$ where $k>0$ and $p$ is a prime number.

So suppose there is such a group. Then if we let $n_{p}$ denote the number of $p$-Sylow subgroups of $G$ we have that $n_{p}=p+1$. Now by letting $G$ act on $Sylow_{P}(G)$ by conjugation we obtain a group homomorphism $G \rightarrow S_{p+1}$. Since $G$ is simple then either $ker(f)$ is trivial or all $G$. Now here’s my question: assume $ker(f)=G$ this would imply then that $G$ has a unique $p$-Sylow subgroup no? but then such subgroup is normal which contradicts the fact that $G$ is simple. So the map in fact is injective but then $|G|$ divides $(p+1)!$ which cannot be.

Basically my question is if my argument is correct, namely thta if $ker(f)=g$ implies the existence of a unique $p$-Sylow subgroup which implies such subgroup is normal in $G$ which cannot be. In case this is wrong, how do you argue that $ker(f)$ cannot be all $G$?

- Show that $S_4/V$ is isomorphic to $S_3 $, where $V$ is the Klein Four Group.
- Prove that $gNg^{-1} \subseteq N$ iff $gNg^{-1} = N$
- What is $\mathbb Z \oplus \mathbb Z / \langle (2,2) \rangle$ isomorphic to?
- Defining an “additive” group structure on $$
- Finding cosets of $(\mathbb{Z}_2\times \mathbb{Z}_4)/\langle (1,2)\rangle$
- $p$ prime, Group of order $p^n$ is cyclic iff it is an abelian group having a unique subgroup of order $p$

Thanks

- $\varphi$ in $\operatorname{Hom}{(S^1, S^1)}$ are of the form $z^n$
- The direct sum $\oplus$ versus the cartesian product $\times$
- normalized subgroup by another subgroup
- Infinite group must have infinite subgroups.
- Find a finite generating set for $Gl(n,\mathbb{Z})$
- What good are free groups?
- Let $G$ a group of order $6$. Prove that $G \cong \Bbb Z /6 \Bbb Z$ or $G \cong S_3$.
- Different ways of constructing the free group over a set.
- Showing a free abelian group is generated by its basis
- What can we say about functions satisfying $f(a + b) = f(a)f(b) $ for all $a,b\in \mathbb{R}$?

You are (mostly) correct. If ker(f) = G, then the image of G in the symmetric group is just the identity, so G does not move any of its Sylow p-subgroups around. However, G acts transitively on its Sylow p-subgroups (they are all conjugate) and the identity is not transitive unless p+1=1, which is silly.

If ker(f) = 1, then G embeds in the symmetric group on p+1 points, so its order divides (p+1)!. This is not a contradiction when k=1.

Indeed, taking p=2, the non-abelian group of order six has order p(p+1) and has exactly p+1 Sylow p-subgroups, and the homomorphism f is injective.

Of course a group of order p(p+1) is also not simple, but probably for a different reason.

“Now here’s my question: assume ker(f)=G this would imply then that G has a unique p-Sylow subgroup no? “

yes, the group action (conjugation on sylow subgroups) is transitive.

- inverse Laplace transfor by using maple or matlab
- What's the lower bound of the sum $S(n) = \sum_{k=1}^n \prod_{j=1}^k(1-\frac j n)$?
- A bad Cayley–Hamilton theorem proof
- Liouville's theorem for Banach spaces without the Hahn-Banach theorem?
- An arctan integral $\int_0^{\infty } \frac{\arctan(x)}{x \left(x^2+1\right)^5} \, dx$
- When do Sylow subgroups have trivial intersection?
- relation of R-module homomorphisms with direct sums
- Prove XOR is commutative and associative?
- reference for multidimensional gaussian integral
- Let G be a finite group with more than one element. Show that G has an element of prime order
- Linear bound on angles in an euclidean triangle.
- Why does the sign $\times$ vanish in mathematical expressions?
- Proving if $F^{-1} $ is function $\Rightarrow F^{-1}$ is $1-1$?
- Irreducible, finite Markov chains are positive recurrent
- Applications of Gauss sums