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Poirot1
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Assume that G of order 48 has 3 sylow 2-subgroups. Let G act on the set of such subgroups by conjugation. How do I know that this action is onto? I know that all 3 subgroups are conjugate but I'm not sure this is enough.
The image of a group action is the set of all possible outcomes or transformations that can be achieved by applying the group's actions to a given object or set. It represents the range of possible changes that can occur through the group's actions.
The image of a group action is determined by the group's elements and their corresponding actions on the given object or set. Each element of the group will result in a different transformation or outcome, thus contributing to the overall image of the group action.
Yes, the image of a group action can be a subset of the original object or set. This can occur when the group's actions result in repetitive or redundant transformations, leading to a smaller image compared to the original object or set.
The image of a group action can be visualized or represented in various ways, depending on the context and purpose. It can be represented as a set of points or objects, a graph or diagram, or a mathematical function mapping the group's elements to their corresponding transformations.
Studying the image of a group action can provide insights into the group's structure and properties, as well as the behavior of the given object or set under the group's actions. It also has applications in various fields such as geometry, physics, and computer science.