Some GE built washing machines use a piece called a air bell coupling to connect the agitator to the transmission. To access this part you need to remove the agitator, to do this you need to grab the base of the agitator and firmly pull up on it. There are no bolts/nuts holding it down. Once the agitator has been removed you will see the air bell coupler, there is a single bolt on the top of it to remove it. With the air bell coupling removed take a close look at the splines on the inside of it, the splines should go ALL the way through. If any of it is stripped out, then it needs to be replaced.
Notice in the picture above that the splines only go for about 1/8″ before it strips out. Past that it is all rounded out on the inside. That is what will cause the agitator not to move.