Second center

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This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
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Definition with symbols

The second center of a group G, denoted Z_2(G), is defined in the following equivalent ways:

  1. It is the subgroup H of G such that H contains the center Z(G) of G, and H/Z(G) is the center of the quotient group G/Z(G).
  2. It is the set of all elements h \in G such that conjugation by h commutes with conjugation by g for every g \in G. In other words, it is the subgroup comprising the elements whose induced inner automorphisms centralize all inner automorphisms.
  3. It is the second member of the upper central series of G.

For more about the properties satisfied and not satisfied by this, see upper central series.