# The Lengths of Certain Real Conjugacy Classes and the Related Prime Graph
## 3 The Lengths of Real Classes  
In ([10], Theorem A), [PERSON] proved that \(G\) is solvable if all noncentral real classes of \(G\) have prime size, and, furthermore, \(\mathsf{cs}_{r}(G)\) is contained in a set of the kind \(\{1,2\}\), \(\{1,p\}\) or\(\{1,2,p\}\), where \(p\) is an odd prime. We show that similar conclusions are true under more general conditions. We begin with Theorem 1, and a basic result of this section which is the reasoning behind several other results. It shows that the properties of certain real elements can determine those of all the real elements in some ways:  
**Theorem 1**. : _(1) Every \(g\in\operatorname{Re}(G)\) is a \(2\)-element if and only if every \(g\in\operatorname{Rp}(G)\) is a \(2\)-element._  
_(2) Let \(p\) be an odd prime, then \(\operatorname{cs}_{r}(G)=\{1,p\}\) if and only if \(\operatorname{cs}_{r}^{*}(G)=\{1,p\}\)._  
_(3) Let \(p\) be an odd prime, then \(\operatorname{cs}_{r}(G)=\{1,2,p\}\) if and only if \(\operatorname{cs}_{r}^{*}(G)=\{1,2,p\}\)._  
Proof. : (1) Necessity being clear. Assume that every primary real element of \(G\) is a \(2\)-element, and let \(g\in\operatorname{Re}(G)\). We can factor \(g\) as \(g=g_{1}g_{2}\cdots g_{n}\), where each element \(g_{i}\) has distinct prime power orders and commute pairwise. By Lemma 2(b), all \(g_{i}\)'s are real in \(G\), and therefore, by hypothesis, they are all \(2\)-elements. Thus, \(g\) is a \(2\)-element, as required. (2) First, assume that \(\operatorname{cs}_{r}(G)=\{1,p\}\). Let \(g\in\operatorname{Re}(G)\backslash\operatorname{Re}^{*}(G)\), then we can write \(g=g_{1}g_{2}\cdots g_{n}\) where each element \(g_{i}\) have distinct prime power orders and commute pairwise. If \(\operatorname{cs}_{r}^{*}(G)=\{1\}\), then all the \(g_{i}\)'s are central in \(G\), so \(g\) is central in \(G\) and \(|g^{G}|=1\). We conclude that \(\operatorname{cs}_{r}(G)=\{1\}\), a contradiction. If \(\operatorname{cs}_{r}^{*}(G)=\{p\}\), then, by Lemma 3(b), it is easy to get \(p||g^{G}|\) and so \(g\
otin Z(G)\). Since \(g\) is arbitrary, we get \(\operatorname{cs}_{r}(G)=\{p\}\), a contradiction too. Thus, \(\operatorname{cs}_{r}^{*}(G)=\{1,p\}\). Conversely, assume that \(\operatorname{cs}_{r}^{*}(G)=\{1,p\}\). For every \(g\in\operatorname{Rp}(G)\), we have \(|g^{G}|=1\) or \(p\), in each case, by Lemma 2(a), we have \(|g|=2\). Now, by (1), we see that every \(g\in\operatorname{Re}(G)\) is a \(2\)-element. By assumption \(|g^{G}|=1\) or \(p\), and therefore \(\operatorname{cs}_{r}(G)=\{1,p\}\), the claim follows. (3) Assume that \(\operatorname{cs}_{r}(G)=\{1,2,p\}\), by Lemma 7 we get \(G=H\times O\) where \(H\) is a dihedral group of order \(2p\) and \(\operatorname{cs}_{r}(O)=\{1\}\). Notice that, for \(g=ho\in G=H\times O\), we have \(|g^{G}|=|h^{H}|\cdot|o^{O}|\), and \(\operatorname{cs}_{r}^{*}(H)=\{1,2,p\}\), and it follows that \(\operatorname{cs}_{r}^{*}(G)=\{1,2,p\}\). Conversely, assume that \(\operatorname{cs}_{r}^{*}(G)=\{1,2,p\}\). If no nontrivial primary real element of \(G\) has an odd order, then it is easy to see that every real element has an even order. Lemma 5 yields that \(G\) has a normal Sylow \(2\)-subgroup \(D\), and \(\operatorname{Re}(G)\subseteq D\) by Lemma 2(d). That is, \(\operatorname{Re}(G)\subseteq\operatorname{Re}^{*}(G)\), and the claim follows. Therefore, we may assume that there exists a nontrivial primary real element \(x\) of \(G\) having odd order, and clearly \(|g^{G}|=2\). Let \(M=\mathbb{C}_{G}(x)\). By Lemma 6, for every \(1\
eq y\in\langle x\rangle\), \(\mathbb{C}_{G}(y)=M\) has index \(2\) in \(G\). Choose element \(z\in\operatorname{Re}^{*}(G)\) such that \(|z^{G}|=p\). Observe that \(z\) is an involution by Lemma 2(a). Now, if any \(1\
eq y\in\langle x\rangle\) centralizes \(z\), then, by Lemma 3, it is easy to see that \(yz\in\operatorname{Re}^{*}(G)\) and \(|(yz)^{G}|=2p\). This is a contradiction by hypothesis. Therefore, \(C_{G}(z)\cap\langle x\rangle=1\). In particular, \(z\
otin M\) and hence \(\langle z\rangle\) is a complement of \(M\) in \(G\).