Group independent linear threshold secret sharing refers to a t out of n linear threshold secret sharing scheme which can be used with any finite Abelian group. A formal definition of a group independent linear threshold sharing is developed. Further, we provide lower bounds concerning the rank, the amount of randomness required and the number of subshares needed for a group independent linear threshold sharing scheme. Lastly, we discuss the group independent linear threshold sharing scheme developed by Desmedt and Frankel. We introduce new algorithms which will reduce the number of required arithmetic operations and group operations needed for the Desmedt-Frankel scheme.