Functional subgroups are specific categories or subdivisions within a larger organization or system that focus on a particular function or area of expertise. These subgroups are typically formed to enhance efficiency, streamline processes, and improve specialization in various tasks and responsibilities. For example, in a corporate setting, functional subgroups could include: 1. **Human Resources** - Focused on recruitment, employee relations, training, and benefits management.

Physicists at the University of Cambridge are researchers, academics, and students engaged in the study of physics within one of the world’s leading universities. The University of Cambridge has a long history of contributions to the field of physics, with many renowned physicists associated with the institution, including Isaac Newton, James Clerk Maxwell, and Stephen Hawking.

The American Association of Physics Teachers (AAPT) is a professional organization dedicated to improving the teaching and learning of physics. It serves educators at all levels—from elementary through university—and focuses on initiatives that promote effective teaching practices, curriculum development, and research in physics education. The presidents of the AAPT are elected officials who serve a term and lead the organization in its mission.

In the context of group theory, an automorphism is an isomorphism from a group to itself. More formally, let \( G \) be a group. An automorphism is a function \( \phi: G \to G \) that satisfies the following properties: 1. **Homomorphism**: For all elements \( a, b \in G \), \( \phi(ab) = \phi(a) \phi(b) \).