Bounding the exponent of the commutator subgroup of a finite p-group

Assume G is a finite p-group. We prove that if G is of nilpotency class c, then \(\exp (\gamma _2(G))\) $exp({\gamma }_2(G))$ divides \(p^{\lceil \log _pc \rceil -1}\exp (G/Z(G))\) $p^{\lceil {log}_{p}c\rceil -1}exp(G/Z(G))$ , and if G is a metabelian p-group of nilpotency class at most \(2p-1\) $2p-1$ , then \(\exp (\gamma _2(G))\) $exp({\gamma }_2(G))$ divides \(\exp (G/Z(G))\) $exp(G/Z(G))$ . Moreover, we prove that \(\exp (H_2(G, \mathbb {Z}))\) $exp(H_2(G,\mathbb{Z}))$ divides \(\exp (G)\) $exp(G)$ if G is a metabelian p-group of nilpotency class at most \(2p-1\) $2p-1$ .