Signing an email as a group

Best,
Group.
Using of article "The" is grammatical not correct.

Nope, you're wrong. It kind of depends on the name of the group. Using a definite article is not necessary incorrect.

Best,

The Bird Watchers of America
The Green Party
The Dog Lovers of Utah
The Friends of Athabasca Library
The Tone-arms Band
The Oshkosh Swamp Lizards Football Club

or

Best,

Tom, Dick, and Harry