Can anyone please tell me The relationship between $p$-solvable Group
and solvable group.and find an example of a $p$-solvable group that is not solvable group or vice-versa.



The relationship between $p$-solvable Group and solvable group
A finite group is solvable if an only if it is $p$-solvable for every prime $p$. An example of a $p$-solvable group which is not solvable is the semidirect product $G = VSL(2,5)$, where $V$ is an elementary Abelian group of order $121$, and $SL(2,5)$ acts faithfully and irreducibly as a group of linear transformations on $V$. The group $G$ is $11$-solvable, but is not solvable.


Later edit: Easier, but perhaps less interesting, is that a non-Abelian finite simple group $G$ is $p$-solvable if and only if the prime $p$ does not divide $|G|$. More generally, any finite group $G$ is $p$-solvable for all but finitely many primes $p$.


(Recall that a finite group $G$ is $p$-solvable for a prime $p$ if and only if every composition factor of $G$ is either of order $p$ or of order prime to $p$.
It is in fact true that a finite group $G$ is solvable if and only if $G$ is $2$-solvable, but this requires the very deep Feit-Thompson Theorem that groups of odd order are solvable).


