Hi,
1) Yes, it is possible for you to buy over your girlfriend's ex's share.
 2) There is a clause mentioning this on the HDB weblink that one divorce party may include another person to retain the flat.
 3) The easiest way is to get legal solemnization and purchase under the Public scheme or alternatively under the fiance/fiancee scheme. You will have to check with the HDB which can speed up the process.
 https://services2.hdb.gov.sg/webapp/BF12AWFeedback/BF12SFrontController?service=ServiceFeedback&operation=createFeedback
4) Both ways can be done even if the flat is still within MOP. Since we can't really control a life event.
Hope the above answer your main concerns, but if there are more query, please feel free to contact me at 
90110636 
, or email: ling.ck7@gmail.com if more information is needed.
 I'll be glad to assist.
Best regards
 Ling CK 
90110636 
 ling.ck7@gmail.com
 https://R056727F.propnex.net/
 https://www.facebook.com/Homesellerbuyer
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