Is there any formula with the help of order of a group to check whether group is abalian or not...?

Question

Is there any formula with the help of order of a group to check whether group is abalian or not...?

Answer 1

There are various property-forcing conditions on group orders.

• Nilpotence: A group of order $n$ is necessarily nilpotent if for every prime power divisor $q\mid n$ and coprime prime divisor $p\mid n$, we have $p\nmid (q-1)$. If this condition is not satisfied, then the group may not be nilpotent.
• Abelian: A group of order $n$ is necessarily abelian if it is cubefree (it has no divisors which are perfect cubes, besides $1$) and the previous condition is satisfied. Otherwise, the group may not be abelian.
• Cyclic: A group of order $n$ is cyclic if $n$ and $\varphi(n)$ are relatively prime, where $\phi$ is Euler's totient function that counts the elements of $(\Bbb Z/n\Bbb Z)^\times$, or equivalently the number of integers between $1$ and $n$ which are coprime to $n$. Otherwise, the group needn't be cyclic.

See Nilpotent Numbers (Pakianathan, Shankar).