Unisingular group

Origin
The notion of unisingular group was first formally defined in the paper Recognizing simplicty of black-box groups and the frequency of $$p$$-singular elements in affine groups by Babai and Shalev.

Roughly speaking, it is the opposite of fixed-point-free group in characteristic $$p$$.

Symbol-free definition
A finite group is said to be unisingular in characteristic $$p$$ (where $$p$$ is a prime number) if its action on any Abelian group of exponent $$p$$ has at least one nontrivial fixed point.