Final answer:
The ulnar nerve is the one that runs in a groove on the dorsum of the medial epicondyle of the humerus.
Step-by-step explanation:
The nerve that lies in a groove on the dorsum of the medial epicondyle is the ulnar nerve. This nerve travels in a groove on the posterior aspect of the medial epicondyle of the humerus, a prominent bony projection on the medial side of the distal humerus. The medial epicondyle serves as an attachment point for the powerful grasping muscles of the anterior forearm, differentiating it in size and robustness from the lateral epicondyle, which is smaller and is the origin for the weaker posterior forearm muscles. The ulnar nerve is one of the systemic nerves of the arm, found between the ulnar and radial nerves, and is responsible for innervating various muscles in the forearm and hand.