# The Lengths of Certain Real Conjugacy Classes and the Related Prime Graph
## 1 Notation and Introduction  
Throughout this paper, the terminology is as follows. All groups are finite and \(G\) refers to a group. By a primary element \(x\) and a primary \(p^{\prime}\)-element \(y\) for a prime number \(p\), we mean that \(|x|=r^{n}\) for an arbitrary prime \(r\) and \(|y|=q^{n}\) for some prime \(q\
eq p\). Recall that an element \(g\) of \(G\) is said to be _real_ in \(G\) if \(g\) and \(g^{-1}\) are conjugate in \(G\), i.e., there is an \(x\in G\) such that \(g^{-1}=g^{x}\), and as usual the conjugacy class of the real element \(g\), \(\{g^{x}|x\in G\}\), is called a real class of \(g\). We denote by \(x^{G}\), \(|x^{G}|\), \(\mathrm{Re}(G)\), and \(\mathrm{Rp}(G)\) the conjugacy class of \(x\in G\), the length of \(x^{G}\), the set of real elements of \(G\), and the set of real primary elements of \(G\), respectively. We denote by \(n(\Sigma)\) the number of connected components of a graph \(\Sigma\). Define the set \(\mathrm{Re}^{*}(G)=\big{\{}x|x\in\mathrm{Re}(G)\big{\}}\) and \(\pi(|x|)\subseteq\{2,p\}\), where \(p\) is an arbitrary prime number\(\big{\}}\). That is, \(\mathrm{Re}^{*}(G)\) is the set of real primary elements and real biprimary even order elements of \(G\). Let \(\mathrm{con}(G)\), \(\mathrm{con}_{G}(G)\) and \(\mathrm{con}_{\tau}^{*}(G)\) denote the set of conjugacy classes, the set of real conjugacy classes, the set of real conjugacy classes \(\{g^{G}\}\), where \(g\in\mathrm{Re}^{*}(G)\), of \(G\), respectively. We denote by \(\mathrm{cs}(G)\), \(\mathrm{cs}_{\tau}(G)\), and \(\mathrm{cs}_{\tau}^{*}(G)\) the set \(\{|C|:C\in\mathrm{con}(G)\}\), \(\{|C|:C\in\mathrm{con}_{\tau}(G)\}\), and \(\{|C|:C\in\mathrm{con}_{\tau}^{*}(G)\}\), respectively. Other notation will be explained in due. The prime graphs are useful tools for intuitively reflecting the structure of groups, and the investigation of them are still underway. The prime graph related to \(\mathrm{con}(G)\), denoted by \(\Gamma(G)\) (introduced by [PERSON] in [1] as \(\Gamma^{*}(G)\)), is a simple graph with vertex set \(\mathrm{V}(G)\) the set of primes dividing the length of some \(C\in\mathrm{con}(G)\) and there is an edge between two vertices \(p\) and \(q\) if and only if the product \(pq\) divides the length of some \(C\in\mathrm{con}(G)\). Prime graphs related to a subset of classes of a group is constructed in the same way. Let \(\Gamma_{r}(G)\) be the prime graph related to \(\mathrm{con}_{r}(G)\) (introduced by [PERSON] in [2] as \(\Gamma^{*}(G)\)). The influence of conjugacy class lengths on the structure of groups has been extensively studied in the past two decades (see [3; 4; 5; 6]). In particular, the problems of recognizing the solvability of \(G\) and the existence of normal (or abelian) \(p\)-complements of \(G\) in this context have been widely investigated, which are also the major themes of our discussion here. Recently, some authors do their research by placing conditions on only real classes of \(G\). Fruitful achievements are obtained in this line of study, and it turns out that the lengths of these classes are closely related to some fundamental properties of \(G\), see, for instance, [2; 7; 8]. The new techniques developed in those papers are valuable and some analogue methods are employed in this article. Now, we mention several notable findings related to real classes. In ([8], Theorem 4.2), [PERSON] proved that, if the class length of every real primary \(2^{\prime}\)-element of \(G\) is either a \(2\)-power or not divisible by \(4\), then \(G\) is solvable. It was shown in ([2], Theorem 6.1) that \(2\) is not a vertex of \(\Gamma_{r}(G)\) if and only if \(G\) has a normal Sylow \(2\)-subgrpup \(S\) (i.e., \(G\) is \(2\)-closed) and \(\operatorname{Re}(S)\subseteq\)Z\((S)\). For odd primes, the outcomes are not that satisfactory. Furthermore, it follows from ([9], Theorem B) that, if the prime graph \(\Gamma_{r}(G)\) of \(G\) is disconnected, then \(2\) must be a vertex of \(\Gamma_{r}(G)\).