Potassium has one more electron than Argon, and that is enough to bump that electron to the 4th level orbital shell. Potassium now differs from the previous period because it uses a 4th orbital shell. Yet it also has something in common with its group members...its valence electron number!
Remember, valence electron number is NOT the same as the total number of electrons for that element. For example, Sodium has a total of 11 electrons, but only 1 valence electron. Likewise, Cesium has only 1 valence electron, but in total has 55 electrons! Their commonality is having only 1 valence electron, and hence they react the same way chemically. And THAT is why they are group buddies.