Smaller groups {homomorphism, group} can have same elements and same multiplication table as larger-group subgroup. For example, C4v group is homomorphic to group of multiplications of 1 and -1. Several extended-group members can associate with same smaller-group member {many-to-one mapping}. Isomorphism and homomorphism are independent group properties.
Two groups can have same number of elements and same multiplication table {isomorphism, group}. For example, group of eight 2x2 matrices with elements 0 and 1 is isomorphic to C4v group of square symmetries. Isomorphism and homomorphism are independent group properties.
3-Group Theory-Mathematical Groups
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Date Modified: 2022.0225