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charlamov
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show that H is subgroup of finite index in Z^n exactly when H is free comutative group of rank n
charlamov said:show that H is subgroup of finite index in Z^n exactly when H is free comutative group of rank n
charlamov said:show that H is subgroup of finite index in Z^n exactly when H is free comutative group of rank n
A subgroup of finite index is a subset of a larger group that contains a finite number of elements and has a finite number of cosets, or distinct left or right translates.
The index of a subgroup is determined by dividing the order of the larger group by the order of the subgroup. This gives the number of cosets, or distinct left or right translates, of the subgroup within the larger group.
A subgroup of finite index has several important properties, including being normal and having finite cohomological dimension. This makes it a useful tool in studying the structure and properties of a larger group.
Yes, a subgroup of finite index can have infinite order. The index of a subgroup only determines the number of cosets, not the order of the elements within the subgroup.
The index of a subgroup can give insight into the structure and properties of the larger group. A subgroup with a small index may be more influential in the overall behavior of the group, while a subgroup with a larger index may have less impact.