Learn Italian: Non dargliela! Ha già comprato una calcolatrice. - Do not give it to him! He already bought a calculator.

For the negative informal imperative (telling someone you call 'tu' not to do something), Italian uses non followed by the infinitive of the verb. 'Don't give' is non dare. When attaching pronouns to the end of an infinitive, you drop the final -e, which is how dare + gliela becomes dargliela.

How does gliela break down in this sentence?

It is a combined double pronoun. Gli means 'to him' (the indirect object) and la means 'it' (the direct object, replacing the feminine noun calcolatrice). When gli is followed by la, it changes to glie- to make pronunciation smoother, forming gliela.

Since this is a negative imperative, can I put the pronouns before the verb instead?

Yes! One of the special rules for the negative informal imperative is that you can choose where to put the pronouns. You can attach them to the end of the infinitive (Non dargliela!) or place them before the verb (Non gliela dare!). Both options are completely correct and common.

Why is già (already) placed in the middle of ha comprato?