It is the man who has to give the Get, in order to dissolve the marriage.
Halachicly, if a man remarries without giving a Get to his 1st wife, the 2nd wife is not considered a bigamist, and his children from the 2nd wife are not Mamzayrim.
However, if C”V a woman remarries without a Get from her 1st husband, she’s still considered married to her 1st husband, and any children from the 2nd marriage are Mamzayrim.