It tends to be something that GM's and players alike playing tabletop RPG's tend to avoid like the Rakghoul plague: Splitting the party.
There are numerous reasons why this is considered taboo, both between player's and their GM's and I'm going to brush upon most of these in turn as this past month I've had deal with a few of them and I just wanted to post my thought's.
From the GM's standpoint this can cause balance issues; primarily with set encounters and degrees of appropriate challenge. A simple example would be an encounter geared towards a party of X PC's is stumbled upon by a group half that number or less. Aside from the obvious concern of it being more challenging and potentially dangerous for the PC's (which can easily be corrected by a GM who can scale the encounter on the fly, we'll get more into that later.) the PC's who overcome the challenge on their own are the only ones who get to reap the rewards of XP and/or loot. The loot can be divided amongst the entire party, (provided you have generous PC's) but XP and other non-tangible rewards are a little more tricky. Essentially you can try to balance it out by providing an separate encounter for the other party members with equal rewards sometime in the future but this can lead into our next concern:
Aside from the possible challenge/balance issue that might arise, there is also the concern of keeping your players' included and entertained. Unless any/all of your divided PC's are participating simultaneously in an encounter of their own or are able to join an encounter soon, you run the risk of them getting bored/feeling useless. I've seen that there are three basic obstacles when it comes to getting one or more PC's to join a encounter already in progress: Communication, Distance, & Time. And all three of them intertwine.
Communication: Unless one group of PC's is within sight and/or earshot of another, a means of communication between said groups must be made to establish concerns, alertness, and any pertinent information such as...
Distance: How physically far way one group is from another can be crucial if one group wants to aid the other or wants aid. If the distance is too great, they might not be able to render any assistance. This factor leads into...
Time: Once distance had been established, the amount of time it takes to traverse that distance is crucial. If it takes too long, help not arrive till combat is over, for better or worse.
Now, I'm going to mention a few short examples based on the Dungeon & Dragons and Star Wars Saga Edition games I run.
Ravenloft is a fantasy style game that places greater emphasis upon party Communication. Aside from magic and the like, methods of Communicating over distance in nearly all medieval fantasy games find direct parallels in history. Mounted messengers, lights, smoke signals, drums, homing pigeons, flags, horns, etc. The examples are limitless and though possibly impractical, they can be very useful to savvy players that want to keep in touch. Communication does not need to be limited to near-instantaneous spells and slower primitive methods; it could be something communicated ahead of time. An example being: "If I don't return in 10 minutes, come looking for me." Or "We will meet you at the Crossroads on the 8th Day, else we've run into trouble in the town."
Now, we've established that communication is key, but Distance and Time must be taken into consideration with this information. This has greater importance in more modern or futuristic RPGs. As Communication becomes more prevalent, the distances between communicating parties tend to become greater in turn, a result of the level of technology. PC's in Fantasy RPG's tend to stay within close range of each other for several reasons, one of the being communication difficulties. In Sci-Fi games where the ability to both communicate and traverse immense distances at nearly the speed of light is possible, the limitless size of a galaxy doesn't seem to be really that grand at all.
But don't think that in the future these basic rules don't apply! Communication, Distance, and Time still matter, and now there's a higher chance that though Communication may be established in an instant, the Distance to traverse and the Time it would take to do so might be greater in comparison! Plus if one doesn't have the chance to pick up the Comlink and radio your buds across town, don't expect help to arrive any sooner. ;)
Wow, I've written a lot more than I was expecting I would. So I'm going to split this up into parts. The next part I'll go into how splitting the party isn't necessarily a bad thing if handled correctly.