The effects of detergents on seed germination is a well documented process and seems to ultimately effect the rate of germination. The result that both of you saw was a withering or brown color to the radicals and roots. The radical originates directly from the roots in lettuce plants so it is possible you are both seeing a similar if not the same effect. However, effects of detergents on plants are also dependent on how you treated the plants with the detergent and of course at what concentration.

If you sprayed the leaves of the germinated plant it is likely that the detergent stripped the wax coat off the leaves leaving them exposed to pathogens and making them loose there waterproofing, ultimately killing them (hence the withering and brown look). If you watered the plants or seeds with the detergent solution, then it is likely that the detergent affected the lipid membrane surrounding the root cells. Disrupting this membrane causes the root cells to not be capable of absorbing nutrients and H₂0. The roots will be the first to dye, following by the rest of the plant.

If you want more information there are many web sites discussing this very experiment (Goggle scholar) including peer reviewed articles available at www.pubmed.gov.