GE washer is not agitating
The first thing to check when your washer won't agitate is the belt. Let me tell you how to check the belt & the clutch with out having to take the washer apart.
The belt
Set the timer in the spin cycle and pull the timer knob to turn the washer on.
If the washer drains the water and then spins the way that it should, this tells you that the motor, belt & clutch are not the reason why your GE washer won't agitate.
A broken belt could be the reason your washer is not agitating or spinning.
The agitator
After making sure that the belt, motor & clutch are working ok, lets check the agitator.
The agitator uses a plastic agitator coupler that is couple to the transmission shaft by using splines cut in the coupler and on the transmission shaft.
Too much usage or overloading the washer too many times will wear the agitator coupler splines. With out the splines the coupler could not be coupled to the transmission shaft and that could be the cause why your GE washer is not agitating.
The clutch
A bad clutch could also be the reason the washer won't agitate or spin. Usually when the clutch is bad the washer will be noisy, like when you have bad brakes on your car.
Transmission
The transmission could be the reason your GE washer is not agitating. A common problem with the transmission is oil leaking right on top of the motor pulley and the belt.
As you can see, there could be different reasons why your GE washer is not agitating. You just need to keep checking each part until you find the problem and replace the part responsible for the washer not agitating.