Discussion Overview
The discussion revolves around finding a group \( G \) that contains elements \( a \) and \( b \) such that both \( a^2 = e \) and \( b^2 = e \), while the element \( ab \) has infinite order. The scope includes group theory concepts, particularly focusing on non-abelian groups and the properties of specific matrix representations and rotations.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- Some participants propose that the group cannot be abelian, leading to the exploration of known non-abelian groups such as the symmetries of the equilateral triangle and 2x2 matrices.
- One participant suggests specific 2x2 matrices \( A \) and \( B \) and calculates that \( (AB)^4 = I \), indicating that this combination does not yield an infinite order element.
- Another participant notes that reflections generally have order 2 and questions which matrices would have infinite order.
- There is a discussion about rotations, with one participant suggesting that a rotation by \( \theta \) will have infinite order if \( n\theta \neq 2k\pi \) or \( \theta \neq \frac{2k\pi}{n} \), using \( \theta = \sqrt{2} \) as an example.
- A later reply clarifies that rotations with \( \frac{\theta}{2\pi} \in \mathbb{Q} \) have finite order, while those with \( \frac{\theta}{2\pi} \in \mathbb{R} \backslash \mathbb{Q} \) have infinite order, suggesting a proof may be needed.
- Another participant introduces the infinite dihedral group \( D_\infty \) as an example, providing a specific construction with elements \( a = rs \) and \( b = s \) that satisfies the conditions of the problem.
Areas of Agreement / Disagreement
Participants express differing views on the types of matrices and rotations that can yield an infinite order element. While some agree on the properties of reflections and rotations, the discussion remains unresolved regarding the specific examples that meet the initial criteria.
Contextual Notes
Limitations include the need for further exploration of specific matrix types and rotations, as well as the potential for additional examples beyond those discussed. The mathematical steps and conditions for infinite order elements are not fully resolved.