Is the Group of Units in a Monoid Always Closed Under Its Operation?

[PsychonautQQ]

Homework Statement

Theorm 1: If M is a monoid, the set of M* of all units in M is a group using the operation of M, called the group of units of M.

My question is this always a "real" group? for example, is this 'group' always closed under the binary operation?

Homework Equations

The Attempt at a Solution

 

[micromass]

Homework Statement

Theorm 1: If M is a monoid, the set of M* of all units in M is a group using the operation of M, called the group of units of M.

My question is this always a "real" group? for example, is this 'group' always closed under the binary operation?

Yes. Try to prove it!