# The Lengths of Certain Real Conjugacy Classes and the Related Prime Graph
## 3 The Lengths of Real Classes
Furthermore, as \(\langle x\rangle\cap\mathbb{C}_{M}(z)=1\) and \(|M:\mathbb{C}_{M}(z)|\big{|}|G:\mathbb{C}_{G}(z)|=p\), it follows that \(M=\langle x\rangle\times\mathbb{C}_{M}(z)\) and that \(|x|=p\). In other words, every nontrivial real primary element of \(G\) having an odd order must have order \(p\). Let \(g\in\operatorname{Re}(G)\), write \(g=g_{1}g_{2}\cdots g_{t}\), where all the \(g_{i}\)'s are real in \(G\) and commute pairwise; in addition, \(|g_{1}|=2^{n_{1}}\) and for \(2\leqslant i\leqslant t\), \(|g_{i}|=p_{i}^{n_{i}}\) where \(p_{i}\)'s are odd primes. By the preceding conclusion, we must have \(|g_{i}|=p\) for \(2\leqslant i\leqslant t\). It follows that \(g\in\operatorname{Re}^{*}(G)\), and therefore \(\operatorname{Re}(G)=\operatorname{Re}^{*}(G)\). Thus, assertion \(\operatorname{cs}_{r}(G)=\{1,2,p\}\) follows. With Theorem 1 in mind, we found that some existed results can be restated in forms that are a little more general. For instance, considering Lemma 5 and 7, we have the following two corollaries. Corollary 1 describes the groups with all the real primary elements being \(2\)-elements. **Corollary 1**. : _The following are equivalent:_  
_(1) Every element in \(\operatorname{Rp}(G)\) is a \(2\)-element._  
_(2) \(G\) has no nontrivial real primary \(2^{\prime}\)-element._  
_(3) \(G\) has a normal Sylow \(2\)-subgroup._Corollary 2 characterizes the structure of \(G\) such that the lengths of classes \(g^{G}\) for every noncentral \(g\in\operatorname{Re}^{*}(G)\) is either 2 or \(p\), where \(p\) is a fixed odd prime. Corollary 2: _Let \(p\) be an odd prime. Then, \(\operatorname{cs}_{\tau}^{*}(G)=\{1,2,p\}\) if and only if \(G=H\times O\) where \(H\) is a dihedral group of order \(2p\) and \(\operatorname{cs}_{\tau}(O)=\{1\}\)._  
Theorem 6.1 in [2] states that all real classes of \(G\) have odd lengths if and only if \(G\) has a normal Sylow 2-subgroup \(P\) and \(\operatorname{Re}(P)\subseteq\operatorname{Z}(P)\). Here is a similar result. Theorem 2: _Let \(P\) be a Sylow 2-subgroup of \(G\). Then, all classes of elements of \(\operatorname{Rp}(G)\) have odd lengths if and only if \(P\lhd G\) and \(\operatorname{Re}(P)\subseteq\operatorname{Z}(P)\)._  
Proof: By Theorem 6.1 in [2], we need only show that, if all classes of elements of \(\operatorname{Rp}(G)\) have odd lengths, then so do the real classes of \(G\). Let the class lengths of all \(x\in\operatorname{Rp}(G)\) are odd numbers. We then have \(x^{2}=1\) for all \(x\in\operatorname{Rp}(G)\), and it is easily seen that all the elements of \(\operatorname{Re}(G)\) must be 2-elements. The claim follows by assumption. The next theorem is another basic result of Section 3, and arguments analogous to those of [10] are used in the proof. Theorem 3: _Assume that \(|g^{G}|\) is either a 2-power or an odd prime for every \(g\in\operatorname{Re}^{*}(G)\). Then, \(G\) is solvable, and there exists an odd prime \(p\) such that \(|g^{G}|\) is either a 2-power or \(p\) for every \(g\in\operatorname{Re}^{*}(G)\)._  
Proof: By ([8], Theorem 4.2), we know that \(G\) is solvable. Assume first that \(G\) has a normal Sylow 2-subgroup \(D\), and set \(U=\langle g\in\operatorname{Re}(G):|g^{G}|\) is odd\(\rangle\). Observe that, by Lemma 2(a), \(U\) is an elementary abelian 2-subgroup which is a subgroup of \(Z(D)\). Moreover, we have \(U\trianglelefteq G\). Let \(A=G/C_{G}(U)\). Then, \(|A|\) is odd and, by [PERSON]'s Lemma (see ([12], Theorem 4.34), \(U=V\times\operatorname{C}_{U}(A)\), where \(V=[A,U]\). Note that \(A\) acts faithfully on \(V\) and that \(\operatorname{C}_{V}(A)=\operatorname{C}_{U}(A)\cap V=1\). Every nontrivial \(x\in V\) is an involution, so \(x\in\operatorname{Re}^{*}(G)\) and \(|x^{G}|=|G:\operatorname{C}_{G}(x)|=|A:\operatorname{C}_{A}(x)|\) is either a power of 2 or an odd prime by hypothesis. However, \(2\
mid|x^{G}|\) since \(x\in Z(D)\), we have \(|A:\operatorname{C}_{A}(x)|\) is an odd prime. Then, by Lemma 4, there exists an odd prime \(p\) such that \(|A:\operatorname{C}_{A}(x)|=p\) for every nontrivial \(x\in V\).