There were four Rimmer brothers: John, Howard, Frank and Arnold. Their social father was obsessed with getting them into the space corps and continually tested and cajoled his sons to that end. To ensure they met the height requirements, Mr. Rimmer used a traction table to stretch his sons. By the time Frank was eleven, he was 6 ft 5 inches tall (RD: Better Than Life).
The three older brothers bullied Arnold fiercely, although he saw it as playful high jinks. Even when they placed a landmine in his sandpit, he laughed it off. During their "Three Musketeers" game, Arnold always had to play the Queen of Spain (RD: Polymorph).
In adulthood, Frank bore a strong resemblance to Arnold, except that Frank had a thin moustache and a deeper voice. Lister used a timeslide to visit Frank's wedding to Janine, where Frank repeatedly told him to move aside as they took the wedding photographs. Lister was so surprised at being sent back in time that Frank had to punch him several times before he agreed to leave (RD: Timeslides).
Behind the scenes
Frank was played by an unnamed child actor in "Polymorph" and by Chris Barrie (with a moustache and a deeper voice) in "Timeslides".
In a deleted scene from "Rimmerworld", Kryten revealed that all 3 of Arnold's brothers became highly decorated officers, but all went insane and in some way killed their crews. Arnold was rather touchy on the subject.
In the booklet accompanying the DVD of Series III, it was suggested that Frank and his brothers were actually the sons of the man they called Uncle Frank. Either way, they would actually be Arnold's half brother, as it is revealed in "The Beginning" that his real father was the family gardener "Dungo" Dennis.