📂Abstract Algebra

Socks-Shoes Property: The Inverse of ab is Equal to the Product of the Inverse of b and the Inverse of a

Theorem ¹

For any element $a,b$ of a group $G$, it follows that $(ab)^{-1}=b^{-1}a^{-1}$.

Proof

Since $(ab)^{-1}$ is the inverse of $ab$, $$ab(ab)^{-1}=e$$ multiplying both sides by $a^{-1}$ gives $$b(ab)^{-1}=a^{-1}e=a^{-1}$$ then multiplying both sides by $b^{-1}$ gives $$(ab)^{-1}=b^{-1}a^{-1}$$

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Explanation

This theorem is referred to as the Socks-Shoes Property, which is an analogy to the process of putting on socks and then shoes. If putting on socks is represented by $a$, and putting on shoes by $b$, where barefoot is represented by $e$, then to return to barefoot after putting on socks and shoes in order, one must “first remove the shoes” and then the socks. Mathematically, this can be expressed as follows. $$(ab)^{-1}=b^{-1}a^{-1}$$