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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.

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up vote 3 down vote accepted

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 ;-)

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Then what can you say about the group $A_5$, which is a group of order 60, and 60 is not prime.Still we know that $A_5$ is a simple group. – analysis89 Apr 22 '13 at 7:06
Yes, but there are several groups of order 60 which are not simple- $\mathbb{Z}/60$, $\mathbb{Z}/30\times\mathbb{Z}/2$, $D_{30}$, etc. The only situation in which knowing $|G|$ means that $G$ is simple is if $|G|$ is prime. If $|G|$ is not prime, you cannot say without further information whether $G$ is simple or not. – KReiser Apr 22 '13 at 7:10
@analysis89, up to isomorphism there are $13$ groups of order $60$, and of these, only $A_{5}$ is simple. So by looking at the order only (that was your question) you cannot tell whether a group is simple. There are cases, though, where the order tells you the group is non-simple - I have added a paragraph in my answer. – Andreas Caranti Apr 22 '13 at 7:11
@ Andreas Caranti, Are the three case which you have described in your answer, the only cases for a group to be non-simple? I mean can you say some more results to check that a group of order n is non-simple? – analysis89 Apr 22 '13 at 7:21
So what you really meant to ask was for which orders we can say for certain that the group is not simple? There are many such orders. For example, any nonabelian simple group must be divisible by at least 3 primes and it must be divisible by 8 or by 12. – Derek Holt Apr 22 '13 at 8:14

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 $\mathbb Z/n\mathbb Z$.

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