A Hall subgroup is a subgroup of a finite group that has an order relatively prime to its index in the group. This means that the order of the Hall subgroup divides the order of the group, and the index (the number of cosets) does not share any common factors with that order. Hall subgroups are important because they allow us to study the structure of groups in relation to their normal subgroups and can play a crucial role in understanding p-groups and their Sylow subgroups.
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