- #1
PsychonautQQ
- 784
- 10
The normal subgroup of a normal subgroup need not be normal in the original group (normalcy is not transitive). Could somebody provide me with an example of where this is the case? Thanks :D
They are simple for ##n > 4## (and three). "Simple in" is a bit of a weird wording.PsychonautQQ said:Small symmetry groups... Is the Alternating symmetry group (A_n) always simple in S_n?
If you consider a whole symmetric group it might be more difficult to prove because you have an additional transposition at hand to get closure under conjugation.If not that's the route I'm going to go... It's always normal because the index in S_n is always going to be two obviously... Anyway if I find a normal subgroup of some A_n maybe it won't be normal in S_n? Do you think this is a smart route to take?
One example of Group Normalcy not being transitive is the following scenario: In a group of friends, Person A is considered normal by Person B, and Person B is considered normal by Person C. However, Person A is not considered normal by Person C. This violates the transitive property of Group Normalcy, where if A is normal to B and B is normal to C, then A should be normal to C as well.
Group Normalcy can heavily influence social dynamics, as it determines who is accepted and valued within a group. When Group Normalcy is not transitive, it can create tension and exclusion within the group as some members may be considered normal by some, but not by others.
Yes, Group Normalcy can change over time as societal norms and values evolve. What may have been considered normal in the past may no longer hold true in the present. As groups and individuals adapt to these changes, the concept of Group Normalcy also shifts.
Group Normalcy can have a strong impact on individual self-perception. When a person is considered normal by their group, they may feel a sense of belonging and acceptance. On the other hand, if a person is not considered normal by their group, they may experience feelings of exclusion and low self-worth.
No, Group Normalcy is not a universal concept as it is heavily influenced by cultural values and norms. What may be considered normal within one group or culture may not hold true in another. It is important to consider the context and cultural background when examining Group Normalcy.