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