How is the note held on the a string a g in a c major chord

I believe you should look into triads for more clarity. A Cmaj chord in its most basic form is just 3 notes. These 3 notes are C-E-G and they form what is known as a triad. A Cmaj Chord is made of notes taken from the C major scale. Hope this helps you find your answer. Take care and have a nice day.