My 2009 equinox would not start it would display service stabilitrak and traction control. I was told it was the battery and bought a new one, one week later same issue at starting it. If we connect a

You have a battery drain somewhere and also a ABS problem. For Battery problem first Charge the battery and perform a Battery drain check as shown on this you tube link:-
- https://www.youtube.com/watch?v=KF1gijj03_0
- https://www.youtube.com/watch?v=zdIKNnwEjIs

You mat have a possible After market Teft Alarm drain.

1. Check for and disconnect any aftermarket systems on the vehicle.

2. Perform the Theft System Relearn.

3. If still unsuccessful, reprogram the Theft Deterrent Module (TDM). This will require the TECH2 and the TIS system.

For Abs problem check following:-
1. Test the Electronic Brake Control Module (EBCM) powers and ground.

2. Confirm the serial data wake up circuit has 12 volts with the ignition in run at the EBCM.

3. Check for loose EBCM terminals.

4. Isolate the EBCM on the data circuit to rule out another module interfering with the EBCM communication.

You will note from above if you do not have 12 volts and a fully charged battery the ABS will malfunction.

I’m happy to assist further over the phone at https://www.6ya.com/expert/uttamjeet_2f3adc18600f8ede

Sounds to me like it is the anti theft... Its real common in GM cars.. Read this over and im pretty sure this will fix the problem.. You can put it in yourself... Ive actually used it before and that was 5 years ago my car still runs great. Do It Yourself GM Security bypass

First off you need to have it checked for any DTC'S - diagnostic trouble codes , an to see if a scan tool will communicate with all the modules on the Vehicle ! If the engine light comes on there is most likely a DTC set ,same with stabilitrak and traction control ! This vehicle has several electronic control modules an they are all on a communication network or serial data bus where they share information ! This vehicles charging system is also computer controlled ! This is one you should take to the dealer or other ASE certified repair facility !
The communication among control modules is performed primarily through the GMLAN high speed serial data circuit and the GMLAN low speed serial data circuits. The modules that need real time communication are attached to the high speed GMLAN network. The body control module (BCM) is the serial data gateway between the networks. The purpose of the gateway is to translate serial data messages between the GMLAN high speed buss and the GMLAN low speed buss. The controller area network (CAN) is another serial data communication network used on this vehicle which is dedicated to the electronic brake control module (EBCM) subsystem for stabilization. Below are more detailed descriptions of the individual networks. The gateway will interact with each network according to that network's transmission protocol. Refer to Body Control System Description and Operation for more information about the gateway.

Could be a couple of things,,,, one of your abs sensors is going out or bad,,, or maybe the charging wire coming off the alternator is coroded or loose