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The study of one-dimensional cellular automata exhibiting group properties is presented. The results show that only a certain class of cellular automata rules exhibit group characteristics based on rule multiplication. However, many other of these automata reveal groups based on permutations of their global states. It is further shown how these groups may be utilized in the design of modulo arithmetic units. The communication properties of cellular automata are observed to map favorably to optimal communication graphs for VLSI layouts. They exploit the implementation medium and properly address the physical limits on computational structures. Comparisons of cellular automata-based modulo arithmetic units with other VLSI algorithms are presented using area-time complexity measures.