Groover graduated from the University of Georgia with a degree in law which helped him reach his political roles. He was a member of the Georgia House of Representatives from 1953–57, 1963–65, 1971–75, and 1983–95.[2][3][4] He supported the 1956 change to the Georgia state flag but in 1993 he acknowledged it was offensive to some and worked for a compromise design.[5] He became infamous when in 1964 during a congressional redistricting bill, he hung from a wall and tried to stop a clock before it signaled the formal end of a session. A photographer captured a photo of the incident, and the photo was printed throughout newspapers which made him an embarrassing icon for Georgia politics.[6]
Groover also authored the 1964 law which moved Georgia elections from plurality to the two-round system, in response to the end of the county unit system and the feared rise of African-American voting power in the state.[7] Later he admitted to federal investigators that this means to dilute Black voting power has been racially motivated and that he has been a segregationist.[8]