Some properties of product and self-product automata are analyzed by the product of state transition matrices. It can be shown that the self-product automata of cyclic type automata with 2m+1 (m≧1) states are again cyclic, and the relationships bitween self-product automata are made clear. As the most important result, we show that a set of self-product automata with some states forms a group with the product operation of automata. Moreover, concerning the number of states of automata, the necessary and sufficient condition to form the group can be derived. Furthermore, it can be shown that the sequence of self-product automata with 2^m (m≧1) states converges to the unity type automata.