Simple groups some interesting properties

Question

I have found some interesting results as follows:

1. If $o(G)\not|\ i(H)!$ then $H$ contains a non-trivial normal subgroup of $G$, where $i(H)=[G:H]$.

2.If $o(G)=2m$, where m is an odd prime number then $G$ contains a non-trivial normal subgroup.

I am collecting this kind of property.Actually I want to know that if we can say a group is simple or not by observing its order only. So anyone who knows more generals result please share the result and give some hint to prove those results.

Answer 1

Concerning your question

I want to know that if we can say a group is simple or not by observing its order only.

Given any positive integer $n$, among the groups of order $n$ there is definitely the cyclic group of order $n$. So you can tell a group is simple by looking at its order if and only if the order is a prime number.

On the other hand there are some orders to which belong only non-simple groups, for instance prime powers $p^{n}$, with $n > 1$, and products of two prime powers. Or odd numbers ;-)

Answer 2

Just to add some details to the answer of Andreas Caranti which uses basically these two results:

1) A cyclic group is simple if and only if its order is prime

2) If $n$ is a prime number, then the only group (up to isomorphism) of order $n $ is the cyclic group $\mathbb Z/n\mathbb Z$.