There are (at least) four bands called Purgatory.

One is from Limoges in France.Their musical style is Hardcore music. They took their name from the first lyrics in the Slayer's song, Raining Blood: "Trapped in Purgatory".
They have made one album called "Hate And Fear"

There is a german death metal band called Purgatory, founded 1993. Homepage

Another Purgatory is a small band from Encinitas, California. 5 members, 4 guys and 1 girl.
Peteria Wityak - Rhythm guitar - only girl
other members - max/matt/andy/griffin

Purgatory was a band from Cleveland formed in the early 1980s. They were perhaps best known for their 1986 album Tied To The Trax, which received airplay on the old Z Rock.
Members
Jeff Hatrix - Vocals (1984 -1988)
Andy Boyer - Vocals (1988 -1989) [Replaced Hatrix]
Randy Gonce - Guitar (1985 -1986)
Frank Romano - Guitar (1986 -1989) [Replaced Gonce]
Greg Perry - Guitar (1984 -1986) & (1986-1988)
Dave Felton - Guitar (1986) [Replaced Perry for a short time]
Tony Ross - Guitar (1988 -1989) [Replaced Perry]
Mark Alexander - Bass (1984 -1989)
Kenny Easterly - Drums (1984 -1989)

Purgatory was also the monicker used by the band currently called Iced Earth. The name in this case was taken from the Iron Maiden song "Purgatory" from the album "Killers"