# The Lengths of Certain Real Conjugacy Classes and the Related Prime Graph
## 4 The Class Lengths of Primary Real Elements
By Lemma 9(b), there exists a real \(r\)-element \(y_{1}\) of \(G\) such that \(\tilde{y}=\tilde{y}_{1}\). Hence, \(|\tilde{y}^{\overline{G}}|=|\tilde{y_{1}}^{\overline{G}}|||y_{1}^{G}|\); again, \(|\tilde{y}^{\overline{G}}|\) is also a power of \(p\). The claim follows. Now, if \(H>1\), then \(G/H\) has a normal \(p\)-complement by induction, and therefore \(G\) has a normal \(p\)-complement. Based on the above, we may assume that \(H=1\). In particular, \(M=P\lhd G\). Now, let \(K\) be a Sylow \(q\)-subgroup of \(G\). We have that \(K\) acts coprimely on \(P\). We claim that every element of \(P\) is \(K\)-invariant. Let \(x\in P\). Then, \(|G:\mathrm{C}_{G}(x)|\) is a \(p\)-power since \(x\) is a \(p\)-element of \(G\). Therefore, there exists \(y\in P\) such that \(K^{y^{-1}}\subseteq\mathrm{C}_{G}(x)\). Then, \(K\subseteq\mathrm{C}_{G}(x^{y})\), and we deduce that the conjugacy class \(x^{p}=(x^{y})^{p}\) is \(K\)-invariant. It follows that the action of \(K\) on \(P\) induces an action of \(K\) on the classes in \(P\), and, for every class \(C\) in \(P\), \(K\) fixes some element of \(C\), say \(x_{C}\). Set \(J=\left\langle x_{C}\right|C\in\mathrm{con}(P)\right\rangle\). Clearly, \(K\) acts trivially on \(J\). Since \(J\) contains an element from each class in \(P\), we get \(P=\bigcup_{y\in P}J^{y}\). Hence, \(P=J\) and the action of \(K\) on \(P\) is trivial, which implies that \(G\) has a normal \(p\)-complement \(K\). This completes the proof. Remark 1: _When \(p=2\), Theorem 4.1 still holds of course._  
By the inductive method and utilizing the properties of \(\mathbf{P}\)-groups mentioned before, we give a sufficient condition for a \(\mathbf{P}\)-group to be supersolvable. It seems that to date that there is no similar result of this kind. Theorem 4.2: _Let \(G\) be a \(\mathbf{P}\)-group. Suppose that \(8\
mid|x^{G}|\) for every \(2\)-element and every real primary element \(x\in G\). Then, \(G\) is supersolvable._  
Proof: Assume that the theorem is false and let \(G\) be a counterexample. Thus, \(G\) is not supersolvable. If every maximal subgroup of \(G\) has index a prime, then \(G\) is supersolvable. Hence, \(G\) has a maximal subgroup \(M\) such that \(|G:M|=4\). Therefore, \(|G|\) is even. Since \(G\) is a \(\mathbf{P}\)-group, \(G\) is solvable by the preceding remarks. There exists a minimal normal subgroup \(N\) of \(G\) such that \(N\) is elementary abelian of order \(p^{m}\) where \(p\) is a prime. Consider \(\overline{G}=G/N\). It can be shown that \(\overline{G}\) is a \(\mathbf{P}\)-group. We may suppose that \(n>1\), since, otherwise, \(G\) is supersolvable as required. Let \(\tilde{x}\) be a real primary \(2^{\prime}\)-element of \(\overline{G}\), by Lemma 9(b), there exists a real primary \(2^{\prime}\)-element \(x_{1}\) of \(G\) such that \(\tilde{x}=\tilde{x_{1}}\). We see that \(|\tilde{x}^{\overline{G}}|=|\tilde{x_{1}}^{\overline{G}}|||x_{1}^{G}|\), which yields \(8\
mid|\tilde{x}^{\overline{G}}|\). On the other hand, let \(\tilde{y}\) be a \(2\)-element of \(\overline{G}\); by Lemma 9(a), it is easy to deduce that \(8\
mid|\tilde{y}^{\overline{G}}|\). This shows that \(\overline{G}\) inherits our hypothesis and thus is supersolvable. Note that the class of finite supersolvable groups is a saturated formation, hence \(\Phi(G)=1\) and \(N\) is the unique minimal normal subgroup of \(G\). From these facts, we obtain a maximal subgroup \(M\) of \(G\) such that \(G=MN\) and \(M\cap N=1\). Since \(|G:M|\) is a prime or \(4\), we must have \(|N|=4\), that is, \(p=n=2\). Consider the permutation representation of \(G\) on \(M\). We see that \(G\) is isomorphic to a subgroup of \(\mathsf{S}_{4}\). Hence, \(G\cong\mathsf{S}_{4}\) or \(G\cong\mathsf{A}_{4}\) since \(G\) is not supersolvable. Furthermore, it is easy to check that \(\mathsf{A}_{4}\) is not a \(\mathbf{P}\)-group, so \(G\cong\mathsf{S}_{4}\). We remark that every element of \(\mathsf{S}_{4}\) is real.