The starter solenoid, also called a starter relay in some manuals, is an electromagnetic throw switch which transfers high amperage from the battery to the starter motor. But you probably knew that...
There are a variety of different solenoids used on motorcycles but most of them work in a similar way. When power and ground are supplied to the lighter primary wires a clicking sound can be heard from the solenoid. At that point the heavier gauge battery and starter cables, atttached to the solenoid studs, are internally connected to send power down to the starter motor.
In most Japanese bikes the solenoid has 2 primary wires (+ and -), and the positive wire receives current all the time the ignition switch is on. Before the solenoid can activate the starter though it must have a connection to the negative ground of the frame. This happens when the starter button is pushed. Knowing this allows you to check for both power and ground going to the solenoid under the right conditions (key on or button pushed). To check you would disconnect the two primary wires and use a test light.
This general advice may not apply to your bike. Check your workshop manual to be sure and use it as the final authority on any work you perform.
If testing for power and ground at the solenoid shows they are present, but your starter relay fails to click or activate the starter, then you would test further to see if the solenoid is working.
This can be done by simply testing for power with a test light at the solenoid stud closest to the starter. When the starter button is pushed the solenoid should click and the stud should then have power. If this test fails the solenoid is defective and should be replaced.
If power is present at the starter cable, and if the battery is fully charged, then the problem is either a bad cable or connection, or the starter motor itself.
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