Higher Centers
Quotienting out by the center of a group yields a sequence of groups called the upper central series:
The kernel of the map is the ith center of G (second center, third center, etc.), and is denoted Concretely, the -st center are the terms that commute with all elements up to an element of the ith center. Following this definition, one can define the 0th center of a group to be the identity subgroup. This can be continued to transfinite ordinals by transfinite induction; the union of all the higher centers is called the hypercenter.
The ascending chain of subgroups
stabilizes at i (equivalently, ) if and only if is centerless.
Read more about this topic: Center (group Theory)
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