# Computing quotients of group by elements of its lower exponent$-p$ central series

Let $$G$$ be a finite $$p−$$group of number of generators $$d$$ and exponent$$−p$$ class $$c$$, that is $$c$$ is the smallest integer satisfying $$P_c(G)=1$$ in the series $$G=P_0(G)≥...≥P_{i−1}(G)≥P_i(G)≥...$$ Where $$P_i(G)=[P_{i−1}(G),G]P_{i−1}(G)^p$$.

1/ Can you show me how to calculate $$G/P_i(G)$$´s using GAP system?

2/ Can you show me how to compute $$G/P_1(G)$$ using abelianisation and row-echelonisation (by hand)?

The series is in general called the $$p$$-central series, so the GAP command is PCentralSeries.