# The Lengths of Certain Real Conjugacy Classes and the Related Prime Graph
## 5 The Prime Graph  
The implications of the class lengths of \(G\) on the structure of \(G\) can be studied via prime graphs. Interestingly, prime graphs related to different subsets of classes exhibit some similarities in their graphic features. For instance, it was shown that \(n(\Gamma(G))\leqslant 2\) and \(n(\Gamma_{r}(G))\leqslant 2\) for any group \(G\) (see [1,2]). In addition, \(\Gamma_{p}^{\prime}(G)\leqslant 2\) if \(G\) is a \(p\)-solvable group, where \(\Gamma_{p}^{\prime}(G)\) is the prime graph related to the set of classes of \(p^{\prime}\)-elements of \(G\) (see [20]). By the proof of Theorem 2, the condition that all the elements of \(\operatorname{Rp}(G)\) have odd number class lengths is equivalent to saying that all the elements of \(\operatorname{Re}(G)\), and therefore of \(\operatorname{Re}^{*}(G)\), have odd number class lengths, hence Theorem 2 shows that \(2\) is not a vertex of \(\Gamma_{r}^{*}(G)\) if and only if \(G\) has a normal Sylow \(2\)-subgroup \(P\) and \(\operatorname{Re}(P)\subseteq Z(P)\). Our Theorem 7 determines the number of connected components of the prime graphs \(\Gamma_{r}^{*}(G)\), and the finding is consistent with those regarding \(\Gamma(G)\) and \(\Gamma_{r}(G)\). For convenience, let \(g\in\operatorname{Re}^{*}(G)\) be a non-central element, and we denote by \(\Delta_{G}(g)\) the connected component of \(\Gamma_{r}^{*}(G)\) that contains the set of vertices \(\pi(|g^{G}|)\), and \(\Delta_{G}(q)\) the connected component of \(\Gamma_{r}^{*}(G)\) which \(q\) belongs to, where \(q\) is a prime. Considering \(\Gamma_{r}(G)\), by ([9], Theorem 3.4), we have that \(G\) is solvable if \(\Gamma_{r}(G)\) is disconnected. It is tempting to wonder wether the corresponding conclusion for \(\Gamma_{r}^{*}(G)\) is valid or not. However, we are unable to prove or disprove this yet. Nevertheless, for a very restrictive case, we have Theorem 9, the proof of which is via a series of lemmas and theorems, in particular, Theorem 7 and 8 are employed. **Theorem 7**: _For any group \(G\), \(n(\Gamma_{r}^{*}(G))\leqslant 2\)._  
**Proof. ** Working by contradiction, let \(G\) be a group of minimal order such that \(n(\Gamma_{r}^{*}(G)\geqslant 3\). Assume first that there exists a nontrivial element of odd order \(x\in\operatorname{Re}^{*}(G)\). By taking a suitable power, we can assume that \(|x|=p\) with \(p\) an odd prime. By Lemma 2(a), we have \(\Delta_{G}(x)=\Delta_{G}(2)\). Since \(n(\Gamma_{r}^{*}(G))\geqslant 3\), there is a connected component \(\Delta\) of \(\Gamma_{r}^{*}(G)\) with \(\Delta\
eq\Delta_{G}(x)\) and \(\Delta\
eq\Delta_{G}(p)\). Choose \(y\in\operatorname{Re}^{*}(G)\) such that \(\Delta=\Delta_{G}(y)\). Because \(2\
otin\Delta\), \(y\) is an involution. Furthermore, as \(p\
otin\Delta\), \(y\) commutes with a Sylow \(p\)-subgroup of \(G\) and, up to conjugation, we can assume that \(x\) and \(y\) commute. Note that \(xy\in\operatorname{Re}^{*}(G)\), so, by Lemma 3, we get the contradiction \(\Delta_{G}(x)=\Delta_{G}(xy)=\Delta_{G}(y)\). We see that all elements of \(\operatorname{Re}^{*}(G)\) are \(2\)-elements, and so are all elements of \(\operatorname{Re}(G)\). Hence, by Lemma 5, \(G\) has a normal Sylow \(2\)-subgroup \(P\). In particular, \(G\) is solvable by the Feit-Thompson theorem. Let \(N\) be a maximal normal subgroup of \(G\) containing \(P\). Then, \([G:N]=q\) an odd prime and \(\operatorname{Re}(G)=\operatorname{Re}(N)\) by Lemma 2(d). If \(x\in\operatorname{Re}^{*}(G)\), then \(\pi(|x^{N}|)\subseteq\pi(|x^{G}|)\subseteq\pi(|x^{N}|)\cup\{q\}\). By minimality, \(n(\Gamma_{r}^{*}(N))\leqslant 2\), and this implies that a connected component of \(\Gamma_{r}^{*}(G)\) must consist of the single prime \(q\), and \(q\
ot\in\Gamma_{r}^{*}(N)\). Thus, if \(g\in\operatorname{Re}^{*}(G)\) and \(q\in\pi(|g^{G}|)\), then \(|g^{G}|\) is a \(q\)-power and \(g\in\operatorname{Z}(N)\). In addition, by Lemma 2(a), \(g\) is an involution. Let \(Z=Z(P)\lhd G\).