The Integers as a Group

An im­por­tant and fa­mil­iar ex­am­ple of a group is the set of in­te­gers, de­not­ed by $\mathbb{Z}$ (pro­nounced "zee" or "zed"), under the op­er­a­tion of ad­di­tion.

• The set $\mathbb{Z}$ con­sists of all whole num­bers: $\ldots, -2, -1, 0, 1, 2, \ldots$.
• The op­er­a­tion is stan­dard ad­di­tion, de­not­ed by $+$.

This group sat­is­fies all group prop­er­ties:

• Clo­sure: The sum of any two in­te­gers is an in­te­ger.
• As­so­cia­tiv­i­ty: Ad­di­tion of in­te­gers is as­so­cia­tive.
• Iden­ti­ty: The number $0$ acts as the iden­ti­ty el­e­ment, since $a + 0 = 0 + a = a$ for any in­te­ger $a$.
• In­verse: The in­verse of $a$ is $-a$, since $a + (-a) = 0$.