The discussion centers on the theorem stating that if M is a monoid, then the set M* of all units in M forms a group under the operation of M. Participants confirm that this group of units is indeed closed under the binary operation defined in the monoid. The closure property is a fundamental aspect of group theory, affirming that the group of units retains the necessary structure to be classified as a "real" group.
PREREQUISITESMathematics students, particularly those studying abstract algebra, and educators looking to deepen their understanding of group theory and monoid structures.
PsychonautQQ said:Homework Statement
theorem 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?