PsychonautQQ said: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?
A group of units of a set refers to a collection of elements from a set that satisfy certain mathematical properties, such as closure, associativity, identity, and invertibility. These properties allow for the elements to be combined through operations, such as multiplication or addition, and still remain within the set.
Closure in a group of units of a set means that when performing an operation on any two elements within the group, the result will also be within the group. For example, in the group of positive integers under addition, the sum of any two positive integers will always be a positive integer, satisfying the closure property.
The identity element in a group of units of a set is an element that, when combined with any other element in the group, will result in the same element. For example, in the group of real numbers under multiplication, the identity element is 1, as any real number multiplied by 1 will equal itself.
The inverse element in a group of units of a set is an element that, when combined with another element in the group, will result in the identity element. For example, in the group of rational numbers under addition, the inverse of any rational number is the negative of that number, as adding a rational number to its negative will result in 0, the identity element.
A group of units of a set is a subset of a larger set, which satisfies certain mathematical properties. While a set may contain elements that do not have any specific relationship to each other, a group of units of a set is a more structured collection of elements that can be combined in a meaningful way through operations and properties. Additionally, a group of units of a set must have an identity element and inverse elements, while a set does not have these requirements.