# The Lengths of Certain Real Conjugacy Classes and the Related Prime Graph
## 5 The Prime Graph
As \(\Gamma_{r}^{*}(M)\) is a subgraph of \(\Gamma_{r}^{*}(G)\) and \(V_{r}^{*}(M)\cap\pi_{i}\
eq\varnothing\) for each \(i=1,2\), we conclude that \(\Gamma_{r}^{*}(M)\) is disconnected. For (2), let \(N\unlhd G\) with \(|N|\) being odd. By Lemma 12, \(\Gamma_{r}^{*}(G/N)\) is a subgraph of \(\Gamma_{r}^{*}(G)\). In particular, \(\mathrm{V}_{r}^{*}(G/N)\subseteq\mathrm{V}_{r}^{*}(G)=\pi_{1}\cup\pi_{1}\). If \(G/N\) is \(2\)-closed, then \(SN\unlhd G\) is of an odd index, thus \(G=SN\) since \(G=\mathrm{O}^{2^{\prime}}(G)\). In other words, \(G\) is \(2\)-nilpotent, by hypothesis, this is a contradiction. Hence, we may assume that \(G/N\) is not \(2\)-closed. The element \(iN\in G/N\) is clearly an involution. If \(iN\in\mathrm{Z}(G/N)\), we have \(G/N=\mathrm{C}_{G/N}(iN)=\mathrm{C}_{G}(i)N/N\), where the latter equality follows from ([12], Lemma 7.7). Hence, \(SN/N\) is a normal Sylow \(2\)-subgroup of \(G/N\), a contradiction. Therefore, \(iN\
otin Z(G/N)\) and \(|(iN)^{G/N}|>1\), so \(\mathrm{V}_{r}^{*}(G/N)\cap\pi_{2}\
eq\varnothing\). On the other hand, as \(G/N\) is not \(2\)-closed, by Theorem 2, again \(2\in\mathrm{V}_{r}^{*}(G/N)\cap\pi_{1}\). We conclude that \(\Gamma_{r}^{*}(G/N)\) is disconnected. Finally, recall that the famous Feit-Thompson Theorem from 1963 states that every group of odd order is solvable. Therefore, we know that \(G\) is solvable if \(|G|=2r\) where \(r\) is a prime by ([12], Theorem 1.35). This conclusion as well as some arguments similar to those in the proof of ([9], Theorem 3.4) are applied to get the following theorem. **Theorem 9**: _Let \(|G|=2^{n}r\), where \(n\leqslant 2\) and \(r\) is an odd number; if \(\Gamma_{r}^{*}(G)\) is disconnected, then \(G\) is solvable._  
Let \(G\) be a counterexample with minimal order. Then, \(G\) is non-solvable and \(\Gamma_{r}^{*}(G)\) is disconnected. By Theorem 7, we get that \(\Gamma_{r}^{*}(G)\) has exactly two connected components with vertex sets \(\pi_{1}\) and \(\pi_{2}\), respectively. We can assume that \(G\) has no normal Sylow 2-subgroup, since, otherwise, by the Feit-Thompson theorem, \(G\) is solvable. By Corollary 1, \(G\) has a nontrivial real primary \(2^{\prime}\)-element \(x\). Then, \(|x^{G}|\) is even and hence \(2\in V_{r}^{*}(G)\). We assume that \(2\in\pi_{1}\). Therefore, all vertices in \(\pi_{2}\) are odd primes. We divide the remaining proof into three parts. (1) \(G\) is not a non-abelian simple group. By Lemma 11, there exists an involution \(i\in G\) such that \(|i^{G}|>1\) is a \(\pi_{2}\)-number and \(\mathrm{C}_{G}(i)\) has a normal Sylow 2-subgroup, say \(S\). As \(|G:\mathrm{C}_{G}(i)|\) is odd, \(S\) is also a Sylow 2-subgroup of \(G\). Notice that \(\mathrm{C}_{G}(i)\) is solvable. Now, for a real primary \(2^{\prime}\)-element \(x\in G\), \(|x^{G}|\) is even and so it is a \(\pi_{1}\) number. Thus, \((|x^{G}|,|i^{G}|)=1\). It follows from ([21], Theorem 2) that \(G\) is not a non-abelian simple group:  
(2) \(G=O^{2^{\prime}}(G)\) and \(O_{2^{\prime}}(G)=1\). Since \(G\) is not 2-closed, by Theorem 8(1), \(\Gamma_{r}^{*}(O^{2^{\prime}}(G))\) is disconnected. If \(O^{2^{\prime}}(G)<G\), then \(O^{2^{\prime}}(G)\) is solvable by the minimality of \(|G|\). Therefore, \(G\) is solvable since \(G/O^{2^{\prime}}(G)\) is solvable, a contradiction. Thus, \(G=O^{2^{\prime}}(G)\). Since \(G\) is not solvable, \(G=O^{2^{\prime}}(G)\) is not 2-nilpotent. Hence, by Theorem 8(2), \(\Gamma_{r}^{*}(G/O_{2^{\prime}}(G))\) is disconnected. If \(O_{2^{\prime}}(G)\) is nontrivial, then, by the minimality of \(|G|\), we get \(G/O_{2^{\prime}}(G)\) is solvable, and so is \(G\). Thus, we must have \(O_{2^{\prime}}(G)=1\). (3) The final contradiction. Let \(M\) be a maximal normal subgroup of \(G\). If \(|G/M|_{2}=2\), then, by hypothesis, \(|M|_{2}=2\) and \(M\) must be solvable (see ([12], Theorem 1.35)), and so is \(G\), a contradiction. The remaining cases where \(|G/M|_{2}=4\) or \(2\
mid|G/M|\) both contradict (2).