How to add a book item in a group library ?

Bibliorock

For a fresh really new user, how to add a book item in a group library ?

adamsmith

Assuming you have the right permissions (i.e., you are a member of the group and the group allows adding/editing items by members), this is no different from adding items to your personal library: https://www.zotero.org/support/adding_items_to_zotero
If that doesn't help, a more specific question would help.