The discussion revolves around identifying the invertible elements in the group of integers modulo 42, denoted as z42. Participants explore the concept of invertibility in relation to the elements being relatively prime to the order of the group.
The discussion is active, with participants sharing their thoughts and questioning the generality of the concept of invertibility. A hint referencing Fermat's little theorem has been introduced, suggesting a deeper exploration of the reasoning behind the relationship between invertibility and relative primality.
Participants are considering the implications of their findings in the context of group theory, specifically regarding the properties of invertible elements and their relationship to the group's order. There is an ongoing exploration of assumptions related to the definitions and characteristics of groups.
duki said:Homework Statement
which elements of z42 are invertibles?
Homework Equations
The Attempt at a Solution
In my notes I have that invertibles are relatively prime to the order of the group. So I have
1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41.
Is that the case for all groups?
Can you always take the relatively primes and get the invertibles?Is what the case for all groups?
duki said:Can you always take the relatively primes and get the invertibles?