Do STEM PhD admissions committees check a candidate's Statement of Purpose for plagiarism? Or is that not really possible, given the high volume of applications?  Do professors have a plagiarism-detecting application to launch investigations, if necessary?

What if, for instance, a successful math PhD applicant to University X gives his Statement of Purpose to his friend for him to apply next year at a similar PhD program at University Y?

I am just asking out of curiosity - I have no intentions of plagiarizing :)

Also, there are plenty of admissions-consulting businesses that are around, particularly for business school applications, so I wonder if such services are offered for PhD applications and whether PhD admissions committees have a way to say, "this Statement of Purpose was definitely not written by Candidate Z."
At my institution there is no such practice and I haven't heard of it at other institutions.


  What if, for instance, a successful math PhD applicant to University X gives his Statement of Purpose to his friend for him to apply next year at a similar PhD program at University Y?


It is quite common for people do consult other's statements as a reference, that is why you can find many of those (besides teaching statements, research statements, etc.) on various academic web sites. However, a blatant plagiarism would probably hurt your application more, no matter how good the original statement is. The reason for that is that such documents are usually personally tailored. They are written to highlight the writer's ambitions, goals and plans and make them compatible with the school's (or adviser's) vision. It is very hard to find statements that can fit multiple students, except if they are very generic and thus rather bad. Also, don't forget that the statement is a part of a larger package, which is also tailored to tell a convincing story why a student should be admitted. Part of that package are also face-to-face interviews. All of that taken together makes it undesirable to have one document sticking out, and it will stick out at some point of the process.

Personally, even if I found a student that has plagiarized another's statement, I wouldn't call it plagiarism but rather a disappointing lack of dedication. If you can't be bothered to put together a few paragraphs on what is important to in your future and also find it easy to blend into another's view of theirs, it's a way more serious problem then plagiarism.

Finally, as said before, the statement is only one part of the application, it is by no means the most crucial one. So, even if you "get away" with plagiarizing it, it won't guarantee your acceptance.