“The English people are a great people.” Here "the English people" is plural, and "are" is of course plural.
"A great people" seems to me singular. Why? I know the sentence is right, but I don't think the following sentence is also right: “They are a student.”
Is "Their people are a bad lot” the same structure as "The English people are a great people”?
The noun people has both a PLURAL sense and a SINGULAR sense.
In the PLURAL sense, people is used as the plural of person very frequently. It is a plural count noun and takes a plural verb. It never has an –s ending; it is already plural.
There were 15 people in the elevator.
The Portuguese people have chosen a new president. (Quirk)1
People were dancing in the streets. (American Heritage Dictionary)2
(Occasionally, “persons” is preferred to “people,” as in legal and quasi-legal language: “All persons needing a permit must apply to the Administration Office by 6:00 p.m., October 16th.”)
In sentence (a) above, you could also say: “There were 15 persons in the elevator,” if you are an elevator inspector or store administrator, for example. However, “persons” is not appropriate for Sentences (b) and (c).
Another example of the PLURAL sense—to mean the plural of “person”—is the first appearance of people in Left’s example in the box above:
The English people are a great people. (Left)
In contrast, the SINGULAR sense of people is used to refer to ALL the men, women, and children of a particular tribe, nation, country or ethnic group, speaking of them as a UNIT, and so the phrase a great people is indeed singular. It is a singular count noun. You can say:
They are a great people. (Quirk)
The Japanese are an industrious people. (Quirk)
The English people are a great people. (The second occurrence of people in Left’s sentence.)
In sentences (e), (f), and (g), “They,” “The Japanese,” and “The English” refer to all the individuals of a tribe or a nation or a country or an ethnic group. “They,” “The Japanese,” and “The English”—the plural subjects of the sentence—all take a plural verb.
The predicate nominative—a great people—is, in this case, singular: a great people who, all together, form that one country or one ethnic group or one people. Since it is a predicate nominative, you can use a singular noun to describe the unit. Besides using a people as the predicate nominative, you could also use other nouns such as “a group,” “a team,” or “a class,” as in: “They are a great team! They won the World Cup!”
A people—meaning the unit of all the people in a tribe, nation, country or ethnic group—is a singular count noun. It has this form as a plural count noun: peoples. Thus, you can say:
The native peoples of Central and South America (Collins COBUILD English Dictionary)3
The English-speaking peoples (Quirk)
I have known it to happen among savage peoples. (Longman)4
All the peoples of the world desire peace. (Azar)5
So, the word people, meaning the unit of people formed by a national group, has a meaning different from people, meaning different individuals. People—the unit—is a singular count noun, as you can see from “a great people,” and a plural count noun, as you can see in sentences (h), (i), (j) and (k) directly above. A footnote in Azar adds this clarification: “The word ‘people’ has a final –s only when it is used to refer to ethnic or national groups.”
About your thoughts: You are right—“They are a student” is not a good sentence. It does not make sense to say “They are a student” in the same way as “They are a people.” “They are a people” means that “they”—the individuals referred to—all come together to form a people, a nation, an ethnic group—it is an acceptable English sentence. It’s hard to imagine what “they” would refer to in the sentence “They are a student.”
Your sentence, “Their people are a bad lot” (“lot” functions here as a singular count noun, and means “a group”) seems to be the same structure as “The English people are a great people.”
1 Quirk et al: A Comprehensive Grammar of The English Language. Longman, 1985, p. 303. 2The American Heritage Dictionary of the English Language. Third Edition. Houghton Mifflin Company, 1992, p.1341. 3Collins COBUILD English Dictionary. Harper Collins, 1995, p.1223. 4 Biber et al.: Longman Grammar of Spoken and Written English. Longman, 1999, p. 290. 5 Azar: Understanding and Using English Grammar, 3rd. ed. (Pearson Education, 2002), p.92 (footnote).