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Free G-action refers to a group action where every element of the group has a unique effect on the set being acted upon. In other words, no two elements of the group produce the same result. On the other hand, effective G-action refers to a group action where every element of the group has a non-trivial effect on the set being acted upon. This means that at least one element produces a different result than the identity element.
Free G-action is a special case of effective G-action. In other words, if a group action is free, it is automatically effective. However, the reverse is not true. An effective G-action may or may not be free.
One example of a free G-action is the rotation of a regular polygon by a group of rotational symmetries. Each element of the group (corresponding to a different angle of rotation) produces a unique effect on the polygon. An example of an effective G-action that is not free is the action of the group of reflections on a square. The identity element (no reflection) produces the same result as one of the reflections (reflection over a diagonal).
Free G-action ensures that the group elements act uniquely on the set, so it can be used to partition the set into distinct orbits. This allows for a clear understanding of the structure of the set. On the other hand, effective G-action introduces more symmetry and can make the set more complicated to analyze.
Free G-action and effective G-action are important concepts in group theory, which is a fundamental area of mathematics. They help us understand the structure and behavior of groups, and have applications in various fields such as geometry, physics, and computer science. Additionally, these concepts help us classify and compare different group actions, leading to a deeper understanding of their properties.
