I think, Full Text Indexes are created on default file group, if you don't specify any filegroup.

CREATE FULLTEXT INDEX ON Table(Column) KEY INDEX PK_Name on [Cat_Name]; -- creates on defulat file group

CREATE FULLTEXT INDEX ON Table(Column) KEY INDEX PK_Name on ([Cat_Name], FILEGROUP [FG_Cat])

I don't know how to move Full Text Indexes from one filegroup to another.... without dropping and recreating them

BOL says..

FILEGROUP filegroup_name

Creates the specified full-text index on the specified filegroup. The filegroup must already exist. If the FILEGROUP clause is not specified, the full-text index is placed in the same filegroup as base table or view for a nonpartitioned table or in the primary filegroup for a partitioned table.