# The History of Having Settled To Accomplish Studies

###### Theorem

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

{proof}

Fix an $x\in G$. Observe that

 $\phi(x\ast x^{-1})=\phi(1_{G})=\phi(x)\star\phi(x^{-1})$ (1)

Recall that, for any group homomorphism $\phi\colon G\to K$,

 $\phi(1_{G})=1_{K}$ (2)

In other , homomorphisms preserve identity. 11A proof for that statement is attached to the . It follows from (1) and (2) that

 $\phi(x)\star\phi(x^{-1})=1_{K}$ (3)

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

Title The History of Having Settled To Accomplish Studies TheHistoryOfHavingSettledToAccomplishStudies 2013-11-27 10:59:44 2013-11-27 10:59:44 jacou (1000048) (0) 14 jacou (0) Proof msc 20A05