Intereting Posts

Proving $\binom{2n}{n}\ge\frac{2^{2n-1}}{\sqrt{n}}$
On the Whitney-Graustein theorem and the $h$-principle.
The non-existence of non-principal ultrafilters in ZF
Power series expansion
Expected Value of a Binomial distribution?
About the hyperplane conjecture.
Odds of guessing suit from a deck of cards, with perfect memory
Compute the limit of $\int_{n}^{e^n} xe^{-x^{2016}} dx$ when $n\to\infty$
Is there a limit of cos (n!)?
Localization at a prime ideal is a reduced ring
Series Summation,Convergence
Show that if U is a principal ultrafilter then the canonical inmersion is an isomorphism between $\mathcal A$ and $Ult_{U}\mathcal A$
What is $f_\alpha(x) = \sum\limits_{n\in \mathbb{N}} \frac{n^\alpha}{n!}x^n$?
Are these two Banach spaces isometrically isomorphic?
$\mathbb S_n$ as semidirect product

If $G$ is a group whose order is $p^n$($p$ is prime), then $G$ is solvable.

How am I going to show this?

Any help is appreciated.

Thank you.

- An example of a division ring $D$ that is **not** isomorphic to its opposite ring
- On the order of elements of $GL(2,q)$?
- Prove that: the center of any group is characteristic subgroup .
- Can the square of a proper ideal be equal to the ideal?
- Global dimension of quasi Frobenius ring
- A book for abstract algebra with high school level

- $x^p -x-c$ is irreducible over a field of characteristic $p$ if it has no root in the field
- Prove that for each $\sigma \in Aut(S_n)$ $n \neq 6$. $\sigma(1,2)=(a,b_2),\sigma(1,3)=(a,b_3), …, \sigma(1,n)=(a,b_n)$
- Ring of integers is a PID but not a Euclidean domain
- If $xy$ is a unit, are $x$ and $y$ units?
- Subgroups of a cyclic group and their order.
- Show $F(U) = K((x^q -x)^{q-1})$.
- the automorphism group of a finitely generated group
- Is $x^3-9$ irreducible over the integers mod 31?
- Prove that $\mathbb{C} \ncong \mathbb{C}\oplus\mathbb{C}$
- Prove that $K$ is isomorphic to a subfield of the ring of $n\times n$ matrices over $F$.

Try by induction on the power of $p$. If $n=1$, $G$ is solvable by definition as a cyclic group of prime order.

Suppose that statement is true for all $k\leq n-1$. Suppose $|G|=p^n$. By the class equation, the center $Z(G)$ is nontrivial. So $Z(G)$ is normal in $G$ and abelian, hence solvable.

So either $G/Z(G)$ is a $p$-group of smaller order, or it is trivial.

The key theorem to remember is that if $H\unlhd G$ and $H$ is solvable and $G/H$ is solvable, then $G$ is also solvable. If $|G/Z(G)|< p^n$, then by induction $G/Z(G)$ is solvable, so $G$ is solvable. Otherwise you just have $G=Z(G)$.

- Functions whose derivative is the inverse of that function
- Prove that the numerator of $H_{p-1}$ in reduced form is a multiple of $p$ for $p$ an odd prime
- Why is it not true that $\int_0^{\pi} \sin(x)\; dx = 0$?
- Possible Class equation for a group
- How to prove that the Kronecker delta is the unique isotropic tensor of order 2?
- Find the remainder when $1!$+$2!$+$3!$…$49!$ is divided by 7?
- What is the Möbius Function for graphs?
- Why isn't $\mathbb{RP}^2$ orientable?
- Is this determinant identity known?
- Proof using Taylor's theorem
- Subadditivity of the limit superior
- Double induction
- Problem while calculating Frame Check Sequence (FCS) in Cyclic Redundancy Check (CRC)
- Theorems' names that don't credit the right people
- Factor groups of matrices