This paper examines random walks on a finite group G and finds upper bounds on how long it takes typical random walks supported on $(\log|G|)^a$ elements to get close to uniformly distributed on G. For certain groups, a cutoff phenomenon is shown to exist for these typical random walks. A variation of the upper bound lemma of Diaconis and Shahshahani and some counting arguments related to a group equation are used to get the upper bound. A further example which uses this variation is discussed.
"Enumeration and random random walks on finite groups." Ann. Probab. 24 (2) 987 - 1000, April 1996. https://doi.org/10.1214/aop/1039639374