In this paper, we study group signature scheme for privacy preservation of a continually transmitting mobile vehicle. A moving vehicle transmits and receives different types of messages on the road. Such communication is a threat to the privacy of a vehicle as its communication can be used to link its identity to its physical entity. The change of size and nature of the neighborhood of a moving vehicle changes as new vehicles come in its vicinity and old neighbors become distant. This change might warrant a pseudonym update for privacy preservation. Moreover, malicious vehicles may broadcast false messages in the network using the identity of other vehicles. This necessitates authentication of the broadcast in addition unlinkability of broadcast messages. Further, a vehicle should not be able to deny its communication, if warranted by law. Thus, a vehicle should be able to broadcast non repudiable, non replicable messages while remaining anonymous. To achieve these conflicting goals of privacy and non-repudiation, group signature schemes along with pseudonyms have been proposed. Group Signature schemes provide security, anonymity to the broadcast, and traceability as required in a VANET. The present work focuses on existing group signature schemes to verify their effectiveness and overheads. Moreover, a scheme to combine pseudonyms and group signature scheme has been proposed to enhance the effectiveness and efficiency of group signatures in VANETs.