In a large pot, get some water boiling. Prepare the broccoli by cutting most of the tender stalk and florets into bite sized chunks. Blanch the florets and tender parts of stalk about 3-4 minutes, without over cooking. It should still have a bit of crunch. Drain, and set aside to cool.
Peel and seed a large cucumber, and cut into diced pieces, set aside.
Meanwhile, using your fingers, shred the crabmeat into pieces. leaving a few chunky ones.
In a bowl, combine the crab, cucumber, and broccoli. Add a little of the mayonnaise at a time; you don't want to over do it. It shouldn't be swimming in mayonnaise. Add the salt and pepper to taste. Combine well. Cover and set in refrigerator until ready to serve.