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In the large number of cases the employers of Negroes in common labor, in which most of them are now engaged, assert that there is no hope for advancement of Negroes in their employ because Negroes will not work under foremen of their own color.

what i don't understand~

them : Negroes or the employers?
in their employ: Negroes being employed or Employers employing Negroes?
common labor??

Thanks for the help in advance.
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eunjinnyIn the large number of cases the employers of Negroes in common labor (common labor means the workers - not the managers or supervisors), in which (type of work) most of them (the Negroes) are now engaged, assert that there is no hope for advancement of Negroes in their (the companies') employ because Negroes will not work under foremen (a foreman is a low-level supervisor) of their own color.

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Also incidental to your question, the word "Negro" is not used any longer. I have to assume you are using an older text.
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for advancement of Negroes in their (the companies') employ

I wonder if advancement means that when the company hires them, the company can give them hire position from the start, or advancement being that Negroes have a chance of being advanced in their position in the work ( get a higher position) after they are hired and worked there for some time.

Thanks so much...
The company spokesperson writes that Negroes are neither hired into management positions nor advanced into management positions. The reason that he gives is that the Negro workers refuse to work for a Negro supervisor or manager. If the workers will not work for certain people, then the company cannot promote these people from being workers to being foremen or managers..
By the way, this statement is according to the writing you have posted. It is contrary to my personal experiences.
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 BarbaraPA's reply was promoted to an answer.
I wonder if I could say this meaning in the way I made sentences below.

~There is no hope for advancement of Negros in employing them.
~There is no hope for advancement of Negros in their employing.

instead of Negros in their employ.

Emotion: wink