Battery light put in key turn on all dash lights on start engine lights off after 20 sec. battery light on engine running battery charging 13/15.8 volts .why is  the battery light staying on?

A few questions so I can help. Is this a Suzuki motorbike or a car? What EXACTLY does it do when you turn the key? How did you 'check for electric'? A vehicle battery should have between 12.6volts to 13.2 volts.
When you turn a starter motor over it needs a large amount of volts to power it, anything under 10 volts and it will struggle. Even a fully charged battery can struggle to turn a starter motor if it is old or faulty.
Try this. First look at the battery and check the terminals are clean and tight. Sometimes it can be as easy as that. Turn the key and look and listen carefully. Try to start the engine. Do you get absolutely nothing and no dash lights? Do you get dash lights which go off when you start the bike but you only get a quiet 'Click'? Do you get dash lights which go off and then a louder 'Clunk'? Does the engine turn over but it's very slow and lazy?
Absolutely nothing suggests a dead or totally flat battery and a lazy, slow starter is a low charge in the battery. A Quiet click could be the Starter Relay and a louder 'Clunk' could be the starter solenoid or the starter itself.
Use the comment button below with more information and i'll try to help more.

The most likely reason for ignition power issues is either a flat battery or a fault in the charging system. Do you get dashboard lights when you turn the key to the first and second positions? Do the dash lights dim when you turn the key to position three, (Engine start)? Do you get any sounds when you turn the key like a 'click' or the engine turning slowly?
Here is a simple battery test. Turn the ignition to position 2, (ignition On, engine Off), Turn on your lights and switch to Main/High beam. Stand in front of the car an look at the front lights. If the battery is low they won't be very bright.
Do you have a multimeter? Put the probes directly on the battery terminals with red to the positive and black to the negative. The multimeter set to volts should give you a reading of 12.6 volts or higher for a fully charged battery. Anything at or below 9 volts will not be enough to start the vehicle.
** Keep well clear of any moving engine part!**
If you can get the vehicle started, (Jump leads or 'bump start'), use the same method on the battery. This shows if the alternator is supplying charge to the battery. You should get a higher reading, somewhere around 14 volts.
Comment below if you need anymore help.

You've got a good start measuring the voltage at the battery. Looks like you've got a low voltage problem. Common on the GM side post batteries is battery acid leaking at the positive battery cable post on the battery. I would recommend removing and checking the cables/posts on the battery. If everything is clean, reassemble the battery cables to the battery. If there is excessive wetness or corrosion on EITHER of the cables/posts, you should replace the battery first, and clean the cables. A leaking battery is going to give you more trouble down the road if you just clean it. If the battery connections on the existing battery are clean, charge the battery with a battery charger, or jump start the car. Use your trusty meter and measure voltage at the battery with the engine running. (Also do this if you replace the battery for leaks). You should read in the neighborhood of 13.5 - 14.0 volts if your charging system is working correctly. If you are reading in the low 12 volt range with the engine running, something is wrong with your charging system, and most likely putting a strain on the battery, preventing it from receiving the charge it needs. If your existing battery is not leaking, cables are clean and tight, jump starts okay, reads 13.5-14ish volts with the engine running, but refuses to crank the engine over after the engine runs for about 15-20 minutes, it's time for a new battery.

Maybe it wasn't running long enough to charge up the battery?
Stop trying to jump it, when you are having problems with it dying, take the battery off and have it charged. And/or buy a $20 voltmeter to see what the battery shows. If it says less than 12 volts, it is discharged, and needs charging. A good battery that is not run down should have at least 12.5 volts. A new battery that is charged should have very close to 13 volts, maybe 12.8 volts.
When you have the truck running, check the alternator for charging. You put the voltmeter right on the battery and check it again. This time, with truck running, battery should show at least 13.5 volts. That shows the alternator is working. A good alternator will put out about 13 or 14 volts-the regulator will keep it from going any higher to prevent problems, like a melting battery or electrical parts burning out.
I would say get the charging system working right, then address the problem of dying at stop lights. That could be a carburetor problem, if the alternator is working.

Did you check the battery to see if it is up to normal 12.5 volts ?
Has the check engine light been on while the engine is running ?
Do you have headlights and warning lights on the dash when you turn the key to start ?

The alternator is the problem,replace the alternator to repair this problem,you will be looking for an out put of 13.25,volts to 14.25 volts at the battery when the engine is running.

sounds like the alternator is shot and is not supplying sufficient power to the engine or even enough to charge the battery, try replacing it, and you said the dash light for the alternator is on so its most likely the alternator

Most partstores will test battery altenator and starter for free. Either your battery is dead or your connections are not metal to metal contact, or you starter relay may be shot. You can put key in on position and either test with a voltmeter or use a screwdriver and jump terminals on solenoid to see if starter turns.

sound like alternator quit charging the battery.check battery voltage.it should be 12.5 volts.if not charge the battery.remove and have alternator tested at auto zone or advance auto parts.dont forget to turn off radio when disconnect negative battery cable.

maybe a little corrosion on the connection from your alt? it could cause the higher resistance to make the voltage drop. the higher rpm's increase flow of electricity forcing it through the corrosion... maybe???