The dominant key in a given musical composition is the key whose tonic is a perfect fifth above (or a perfect fourth below) the tonic of the main key of the piece. Put another way, the key whose tonic is the dominant scale degree in the main key.
If, for example, a piece is written in the key of C major, then the key of C is the tonic key. The key of G major is the dominant key since it is based on the dominant note for the key of C major.
In sonata form in major keys, the second subject group is usually in the dominant key. Even with the widest roaming modulations in the development, the dominant key exerts influence and eventually forces a return to the tonic key.
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