**Theorem 1**. : _Let \(G\) be a group. Then \(D_{p}(G)=P[A]\), where \(P\) is the Sylow \(p\)-subgroup and \(A\) is a Hall \(p^{\prime}\)-subgroup of \(D_{p}(G)\). Moreover, \(P=D(P)\) and \(A=N(A)\). In particular, \(\cap_{p||G|}D_{p}(G)=N(G)\)._  
**Theorem 2**. : _Let \(G\) be a group. Then_  
_(1)_ \(D_{p}(G)=1\) _if and only if_ \(C_{G}(G^{\prime}(p))=1\)_;_  
_(2)_ \(O_{p^{\prime}}(D_{p}(G))\leq Z_{\infty}(O^{p}(G))\)_;_  
_(3) if_ \(Z(G^{\prime}(p))=1\)_, then_ \(C_{G}(G^{\prime}(p))=D_{p}(G)\)_._  
**2. Elementary Properties on \(D_{p}(G)\)**  
In this section, we list some elementary properties of \(D_{p}(G)\) that will be used in the proofs of the main results. **Lemma 1**. : _Let \(G\) be a group. Then_  
_(1)_ _If_ \(H\leq G\)_, then_ \(H^{\prime}(p)\leq G^{\prime}(p)\)_;_  
_(2)_ _if_ \(N\trianglelefteq G\) _and_ \(N\leq H\leq G\)_, then_ \((H/N)^{\prime}(p)=H^{\prime}(p)N/N\)_;_  
_(3)_ \(G^{\prime}(p)\) _is nilpotent if and only if_ \((G/\Phi(G))^{\prime}(p)\) _is nilpotent;_  
_(4)_ _if_ \(G=MN\)_, where_ \(M\leq G\) _and_ \(N\trianglelefteq G\)_, then_ \(G^{\prime}(p)\leq M^{\prime}(p)N\)_. In particular,_ \((M\times N)^{\prime}(p)=M^{\prime}(p)\times N^{\prime}(p)\)_._  
**Proof. **  
(1) Let \(H\leq G\). Since \(H/(H\cap O^{p}(G))\cong HO^{p}(G)/O^{p}(G)\leq G/O^{p}(G)\), \(H/(H\cap O^{p}(G))\) is a \(p\)-group and so \(H^{\prime}(p)=H^{\prime}O^{p}(H)\leq H^{\prime}(H\cap O^{p}(G))\leq G^{\prime} O^{p}(G)=G^{\prime}(p)\). (2) Let \(O^{p}(H/N)=R/N\). \(O^{p}(H)N\leq R\) Since \((H/N)/O^{p}(H/N)=(H/N)/(R/N)\cong H/R\). Conversely, \(H/O^{p}(H)N\cong(H/O^{p}(H)/(O^{p}(H)N/O^{p}(H))\) and \(H/O^{p}(H)N\cong(H/N)/(O^{p}(H)N/N)\), so \(R/N\leq O^{p}(H)N/N\). Hence \(O^{p}(H/N)=O^{p}(H)N/N\). Then \((H/N)^{\prime}(p)=(H/N)^{\prime}O^{p}(H/N)\)  
\(=(H^{\prime}N/N)(O^{p}(H)N/N)=H^{\prime}O^{p}(H)N/N=H^{\prime}(p)N/N\). (3) Clearly, \(G^{\prime}(p)\) is nilpotent if and only if \(G^{\prime}(p)\Phi(G)/\Phi(G)\cong G^{\prime}(p)/G^{\prime}(p)\cap\Phi(G)\) is nilpotent. So (3) follows from (2). (4) By \(G^{\prime}\leq M^{\prime}N\) and \(O^{p}(G)\leq O^{p}(M)N\), we get \(G^{\prime}(p)\leq M^{\prime}(p)N\). If \(M\trianglelefteq G\), then \(G^{\prime}(p)\leq N^{\prime}(p)M\). Thus \(G^{\prime}(p)\leq M^{\prime}(p)N\cap N^{\prime}(p)M=M^{\prime}(p)(N\cap N^{ \prime}(p)M)=M^{\prime}(p)N^{\prime}(p)(M\cap N)\). Hence, if \(G=M\times N\), then \(G^{\prime}(p)=M^{\prime}(p)\times N^{\prime}(p)\) by (1). \(\qed\)  
**Proposition 1**. : _Let \(G\) be a group. Then_  
_(1)_ \(N(G)C_{G}(G^{\prime}(p))\leq D_{p}(G)\leq D(G)\)_;_  
_(2)_ \(D_{p}(G)\) _is soluble;_  
_(3) if_ \(M\leq G\)_, then_ \(M\cap D_{p}(G)\leq D_{p}(M)\)_;_  
_(4) if_ \(N\trianglelefteq G\)_, then_ \(D_{p}(G)N/N\leq D_{p}(G/N)\)_;_  
_(5) if_ \(G=A\times B\)_, where_ \(A,B\leq G\) _and_ \((|A|,|B|)=1\)_, then_ \(D_{p}(G)=D_{p}(A)\times D_{p}(B)\)_._  
**Proof. **  
(1) Since \(H^{\prime}\leq H^{\prime}(p)\leq H\) and \(H^{\prime},H^{\prime}(p)\) are characteristic subgroups of \(H\), we have \(N_{G}(H^{\prime})\geq N_{G}(H^{\prime}(p))\geq N_{G}(H)\), that is, \(N(G)\leq D_{p}(G)\leq D(G)\). If \(x\in C_{G}(G^{\prime}(p))\), then \(x\) is a normalizer of \(H^{\prime}(p)\) for all \(H\leq G\) by Lemma 1 (1). Hence \(x\in D_{p}(G)\) and so, \(C_{G}(G^{\prime}(p)\leq D_{p}(G)\). (2) It follows from (1) and \(D(G)\) is soluble in ([9], Proposition 2.4). (3) It is easy to see that \(M\cap D_{p}(G)\leq\bigcap_{H\leq M}N_{M}(H^{\prime}(p))=D_{p}(M)\). (4) If \(x\in D_{p}(G)\), then \(xN\) normalizes \((H/N)^{\prime}(p)\) for all \(H/N\leq G/N\) by Lemma 1 (2). Hence \(D_{p}(G)N/N\leq D_{p}(G/N)\). \(H=(H\cap A)\times(H\cap B)\) for all \(H\leq G\) by the hypotheses. It follows from Lemma 1 (4) that \(H^{\prime}(p)=(H\cap A)^{\prime}(p)\times(H\cap B)^{\prime}(p)\). Hence \[ \begin{array}{rcl}N_{G}(H^{\prime}(p))&=&N_{G}((H\cap A)^{\prime}(p))\cap N_{G }((H\cap B)^{\prime}(p))\\ &=&(N_{A}((H\cap A)^{\prime}(p))\times B)\cap(A\times N_{B}((H\cap B)^{\prime }(p)))\\ &=&N_{A}((H\cap A)^{\prime}(p))\times N_{B}((H\cap B)^{\prime}(p)),\end{array}