The name of the sovereign nation is "the United Kingdom of Great Britain and Northern Ireland". (Before Ireland's independence in 1922, it was the U.K. of Great Britain and Ireland.) "Great Britain" is the name of the large island with England, Scotland (to the north), and Wales (a peninsula in the west). The "British Isles" include Great Britain, Ireland, and some smaller surrounding islands. "England" is a division of the U.K. Even today, the divisions of Great Britain have differing cultural traditions and accents (and pockets of surviving languages) that reflect the different invasions and settlements over the past 2500 years. All the earliest settlers in New England came from England, bringing English customs and language. Colonial New England was part of the "British Empire", a term no longer used.
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