Intereting Posts

Prove that integrable implies bounded
Concepts about limit: $\lim_{x\to \infty}(x-x)$ and $\lim_{x\to \infty}x-\lim_{x\to \infty}x$.
Why do proof authors use natural language sentences to write proofs?
$n^s=(n)_s+f(s)$, what is $f(s)$?
Prove this inequality $\frac{1}{1+a}+\frac{2}{1+a+b}<\sqrt{\frac{1}{a}+\frac{1}{b}}$
Factorial Inequality problem $\left(\frac n2\right)^n > n! > \left(\frac n3\right)^n$
Last few digits of $n^{n^{n^{\cdot^{\cdot^{\cdot^n}}}}}$
How can I show that $\sum\limits_{n=1}^\infty \frac{z^{2^n}}{1-z^{2^{n+1}}}$ is algebraic?
$\lim_{n\to \infty} {a^n-b^n\over a^n+b^n}.$
If $f(x) = \sin \log_e (\frac{\sqrt{4-x^2}}{1-x})$ then find the range of this function.
Compactness theorem equivalent
When can a measurable mapping be factorized?
A sum for stirling numbers Pi, e.
The Instant Tangent
Find integer numbers $x,y>1$ such that $x|3^y+1$ and $y|3^x+1$

If $G$ is a groupe such that $|G|=p^m k$, does $G$ has a subgroup $H$ s.t. $|H|=p^n$ with $n<m$ ?

I know that $G$ has a $p-$sylow subgroup, i.e. a group of order $p^m$.

I also know that $G$ has an element of order $p$ and thus a subgroup of order $p$ (in fact $\left<g\right>$ where $g^p=1$).

- Alternative proofs that $A_5$ is simple
- If H ≤ Z(G) ≤ G, where G is a finite group,Z(G) is its center, and (G:H) = p for some prime p, then G is abelian.
- Infinitely many simple groups with conditions on order?
- On the centres of the dihedral groups
- Number of subgroups of prime order
- Is every element of a finite group a product of elements of prime order?

1) But for $1<n<m$, is there a group of order $p^n$ ?

2) By the way, does all $p-$group (i.e. a group of order $p^n$) are abelian ? (in a solution of an exercise, they use such a property but I’ve never seen such a result).

- A finite $p$-group has normal subgroup of index $p^2$
- Where does the ambiguity in choosing a basis for a Lie algebra come from?
- Proving that a subgroup of a finitely generated abelian group is finitely generated
- Computing the Smith Normal Form
- Product of two cyclic groups is cyclic iff their orders are co-prime
- A question about perfect group
- Proof about cubic $t$-transitive graphs
- f(a) = inverse of a is an isomorphism iff a group G is Abelian
- Find four groups of order 20 not isomorphic to each other.
- Adapting a proof on elements of order 2: from finite groups to infinite groups

Yes there is such a subgroup. For a proof look at Theorem 1 in https://en.wikipedia.org/wiki/Sylow_theorems#Proof_of_the_Sylow_theorems.

To answer your second question: not all $p$-groups are abelian. The simplest counterexample is $Q_8$, the quaternion group. It is, however, the case that all $p$-groups have nontrivial center. This is a nifty consequence of the class equation. For $P$ a group of prime power order $p^m$,

$$ |P| = |Z(P)| + \sum_{i=1}^r[P:C_P(g_i)] $$

where $g_1,\dots,g_r$ are representatives of the non-central conjugacy classes of $P$. Since then $C_P(g_i) \neq P$ by definition, we must have $p$ dividing each term of the sum. And since $p$ divides $|P|$ it must also divide $|Z(P)|$ so the center is nontrivial.

Let $P$ be the $p$-sylow sugroup of $G$ then $|P|=p^{m}$, there existe a normal subgroup $N$ of $P$ of order $p^{m-1}$, also there existe a normal subgroup of N of order $p^{m-2}$, ….

All those groups are subgroups of $G$

- An application of Pigeon Hole Principle
- Proof that $e^{-x} \ge 1-x$
- an analytic function from unit disk to unit disk with two fixed point
- How to understand marginal distribution
- Suppose $\lim \limits_{n \to ∞} a_n=L$. Prove that $\lim\limits_{n \to ∞} \frac{a_1+a_2+\cdots+a_n}{n}=L$
- Proving the congruence $p^{q-1}+q^{p-1} \equiv 1 \pmod{pq}$
- Understanding proof of completeness of $L^{\infty}$
- Cyclically reduced words in free groups
- What does it mean to sample, in measure theoretic terms?
- Johann Bernoulli did not fully understand logarithms?
- Construction of an infinite number type and other ideas
- An outrageous way to derive a Laurent series: why does this work?
- Show that we cannot have a prime triplet of the form $p$, $p + 2$, $p + 4$ for $p >3$
- A closed form for $\int x^nf(x)\mathrm{d}x$
- How do I find an integral basis, given a basis consisting of algebraic integers?