# Motivation for the definition “strongly real element” in group?

Elements in finite groups can splitted in real / non-real elements. Which is quite well-motivated definition: element is called real if all characters take real values on it.
Equivalent requirment is: element is conjugate to its inverse.

Definition: There is stronger property: element is called strongly real if it satisfies the following equivalent conditions:

• It is either the identity element or an involution or can be expressed as a product of two distinct involutions (here an involution means a non-identity element whose square is the identity element).
• It is either the identity element or there is an involution that conjugates it to its inverse.

Question: What is the motivation of that definition ?

Further background: It is known that the number of real conjugacy classes is equal to the number of the real characters. Real characters can be splitted further by e.g. Frobenius-Shur indicator as real/quaternionic characters. However such splitting of characters is NOT reflected by splitting conjugacy classes to strongly real. See e.g. MO Are there “real” vs. “quaternionic” conjugacy classes in finite groups?, MO If all real conjugacy classes are strongly real, then all real irreps are “strongly real”(symmetric), true ?, MO Strongly real elements of odd order in sporadic finite simple groups

Remarks: Real and strongly real elements in finite simple groups have been much studied see e.g. presentation Singh, arXiv:1104.3933, arXiv:0809.4412 , arXiv:1303.6085

• Perhaps the main point of interest here is not what it tells you about representations or about the strongly real conjugacy class itself, but what it tells you about involution classes in the group. – Colin Reid Sep 23 '17 at 6:39
• Your definition of strong reality becomes much simpler if you call the identity element an involution (no need for all those cases). Anyway, as you probably know (but I haven't followed up all your links to check), every element of a Weyl group is strongly real; I think this is original to Carter, and is fundamental in his description of conjugacy classes in Weyl groups. See Lemma 5, p. 5, and Corollary (ii), p. 45 (which refers to §3), of Carter - Conjugacy classes in the Weyl group (MSN). – LSpice Oct 6 '17 at 1:25

The real elements of $G$ are exactly those elements $x$ in $G$ such that $x^g = x^{-1}$ for some element $g \in G$. Also, $x$ is strongly real if it is real and we can find $g$ as above with the additional property that $g^2 = 1$. Thus "strongly real" is a more stringent requirement than merely being real.
The condition that $x$ is real is equivalent to saying that $\chi(x)$ is a real number for all characters $\chi$ of $G$. The above definition of "strongly real" is equivalent to the definition given in the question, but is, I think, cleaner and more concise. The property of being strongly real is not reflected in the character values. Note that $D_8$ and $Q_8$ have identical character tables, but in $D_8$, every element is strongly real, and in $Q_8$, only two elements are strongly real.