Start by cleaning the jars well. Invert and allow to dry completely.
Next prepare the jalapeños pepper. Remove and discard the seeds and veins of the pepper (if you're concerned about too much heat if you're not, leave them in!), chop finely. Mince the garlic.
Combine the sugar, salt, vinegar, water, garlic and pepper into pan on the stove. Turn on high and bring to a boil. Reduce heat and continue at a low boil for two minutes. Remove from the heat and set aside to cool
Clean the radishes, remove the root and greens. Slice thinly. Pack into jars. You'll want to pack the radishes tightly into the jars because as they begin to pickle they will shrink so pack them as tightly as possible.
Pour the cooled liquid into the jars. Seal tightly and refrigerate. I try and wait 1 week before using, but after 24 hours, the pickles begin to get sweet and spicy!