From 1877-1948 they did. Victoria adopted the title "Empress of India." The succeeding monarchs were styled "King-Emperor" which is for instance why you have this signature of Edward VIII as "Edward RI" i.e. Rex Imperator: http://en.wikipedia.org/wiki/King-Emperor#mediaviewer/File:Edwardsig.svg

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The British Empire and the British Monarchs' titles

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What was the de jure relatioship between the Kingdom of Great Britain and the Indian Empire?