Discussion Overview
The discussion revolves around the question of how the quotient group G/Go can act effectively on a set X, given a group G of transformations acting on X and a subgroup Go. Participants explore the definitions and implications of effective actions and the well-defined nature of the quotient group's action on X.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant questions how the quotient group G/Go can act on X, noting that each member of the quotient is a set of transformations and suggesting that Go does not provide a one-to-one correspondence from X to X.
- Another participant proposes that for an element h in G, the notation [h] = hGo is in G/G0, and questions whether the action [h]*x := h*x is well defined.
- A subsequent reply agrees with the concern about well-definedness, suggesting that there may be different functions h1 and h2 such that [h1] = [h2], leading to ambiguity in the action on x.
- Another participant points out that the G-action of elements of Go could be used to show that the action of G/G0 is well defined, hinting at a deeper relationship between the elements of G and Go.
- One participant reflects on the implications of [h1] = [h2] and the existence of an element k in Go such that h1 = h2k, leading to a conclusion about the action on x, but expresses uncertainty about the correctness of their reasoning.
- Another participant acknowledges a mistake in their earlier reasoning regarding the end result of the G/Go action.
- A final comment suggests that if elements of G act trivially, they can be referred to as "the identity," implying a relationship between trivial actions and the identity in the context of the quotient group.
Areas of Agreement / Disagreement
Participants express uncertainty about the well-defined nature of the action of G/Go on X, with some agreeing on the need for clarification while others propose different interpretations. The discussion remains unresolved regarding the effective action of the quotient group.
Contextual Notes
Participants note potential ambiguities in the definitions and relationships between elements of G and Go, as well as the implications of the actions on X. There are unresolved questions about the assumptions underlying the definitions of effective actions and the nature of the quotient group.