The Church of England condemned their call for a stricter reading of scripture. Their thinking was that the Reformation was not complete and called for a "purity" in readings and form of worship. This notion compromised the politics of the Anglican Church and Puritan sects were restricted from governance and society because of their dissenting religious beliefs.

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Persecution as a description seems inaccurate. The beliefs of those associated with Puritanism were admonished more than persecuted. I suspect that word was associated with Puritans by Americans seeking to establish religious roots to the founding of the US which is highly overstated.