The History of Having Settled To Accomplish Studies

Theorem

A group homomorphism preserves inverses elements. That is, for groups $\mathrm{(}G\mathrm{,}\mathrm{\ast }\mathrm{)}$ and $\mathrm{(}K\mathrm{,}\mathrm{\star }\mathrm{)}$, and a homomorphism $\varphi \mathrm{:}G\mathrm{\to }K$, $\varphi \mathit{}\mathrm{(}{x}^{\mathrm{-}\mathrm{1}}\mathrm{)}\mathrm{=}\varphi \mathit{}{\mathrm{(}x\mathrm{)}}^{\mathrm{-}\mathrm{1}}$.

Fix an $x\in G$. Observe that

$$\varphi (x\ast x^{-1})=\varphi (1_G)=\varphi (x)\star \varphi (x^{-1})$$(1)

Recall that, for any group homomorphism $\varphi :G\to K$,

$$\varphi (1_G)=1_K$$(2)

In other , homomorphisms preserve identity. It follows from (1) and (2) that

$$\varphi (x)\star \varphi (x^{-1})=1_K$$(3)

Because the inverse of any group is unique, the only value of $\varphi (x^{-1})$ whose product with $\varphi (x)$ is $1_K$ is, of course, $\varphi {(x)}^{-1}$. Therefore, all group homomorphisms preserve the inverse.