Final answer:
The inverse of the product of group elements a₁, a₂, ..., aₙ is the product of the individual inverses in reverse order: (aₙ)⁻¹ (aₙ−₁)⁻¹ ... (a₂)⁻¹ (a₁)⁻¹.
Step-by-step explanation:
If we have elements a₁, a₂ , ..., aₙ belonging to a group in mathematics, the inverse of the product a₁ a₂ ... aₙ is found by taking the inverse of each element in reverse order. That is, the inverse is (aₙ)⁻¹ (aₙ−₁)⁻¹ ... (a₂)⁻¹ (a₁)⁻¹. This is due to the group property that every element has an inverse and the product of an element and its inverse is the identity element of the group.
The concept of inverses is foundational in group theory, which is a branch of abstract algebra. To "undo" or "invert" the effect of a composite operation, we need to apply each operation's inverse in the opposite order. This is analogous to applying subtraction as the inverse of addition, or division as the inverse of multiplication.