An example of length computation for a group algebra of a noncyclic Abelian group in the modular case

We demonstrate that the technique for calculating the length of two-block matrix algebras, developed by the author earlier, can be used to calculate the lengths of group algebras of Abelian groups. We find the length of the group algebra of a noncyclic Abelian group of order $2p^2 $, where $p> 2$ is a prime number, over a field of characteristic $p$, namely, we prove that the length of this algebra is equal to $3p-2$.