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rulin
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what is the order of normalizer of Sylow p-subgroup of simple groups A_n?
The normalizer of a Sylow p-subgroup of a group G is the largest subgroup of G in which the Sylow p-subgroup is a normal subgroup. In other words, it is the subgroup that contains all elements of G that commute with the elements of the Sylow p-subgroup.
The normalizer of a Sylow p-subgroup is important because it helps us understand the structure of the group and its subgroups. It can also help us identify other important subgroups, such as the centralizer of the Sylow p-subgroup.
The normalizer of a Sylow p-subgroup is closely related to simple groups because in simple groups, the normalizer of any non-trivial proper subgroup is the whole group. This means that in simple groups, the normalizer of a Sylow p-subgroup will always be the whole group or the Sylow p-subgroup itself.
Yes, the normalizer of a Sylow p-subgroup can be a proper subgroup in non-simple groups. However, in simple groups, the normalizer of a Sylow p-subgroup is always the whole group or the Sylow p-subgroup itself.
The normalizer of a Sylow p-subgroup can be used as a tool in the classification of simple groups. By considering the normalizers of the Sylow p-subgroups in a simple group, we can identify certain patterns and relationships that help us classify the group into one of the known families of simple groups.