Scan the jacket of the book for G.P. Putnam's Sons logo. It is a black rectangle with rounded edges containing the words "Putnam EST. 1838." If the book has this logo on its facade, then it is assuredly a G.P. Putnam's Sons book. Whether or not it is a first edition will be determined in subsequent steps.
Turn to the title verso page of the book, which is located on the backside of the title page. This page contains information about the publishers, copyright, and edition of the book. G.P. Putnam's Sons books do not explicitly indicate that they are first editions. Rather, they labeled for their edition only in subsequent editions to the first. G.P. Putnam's Sons books that are first editions will have nothing indicating the book's edition.
Scan the title verso page for what is known as a number line. Books that have been published in the last 20 to 25 years will contain a number line indicating its edition number. A full number line lists the numbers 1 through 10, and indicates that the book is a first edition. If the number line contains the number one, no matter its placement along the line, then you can be sure that the book is a first edition G.P. Putnam's Sons.