Second center: Difference between revisions
(Created page with '{{subgroup-defining function}} ==Definition== ===Definition with symbols=== The '''second center''' of a group <math>G</math>, denoted <math>Z_2(G)</math>, is defined in t…') |
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# It is the set of all elements <math>h \in G</math> such that conjugation by <math>h</math> commutes with conjugation by <math>g</math> for every <math>g \in G</math>. In other words, it is the subgroup comprising the elements whose induced inner automorphisms centralize all [[defining ingredient::inner automorphism]]s. | # It is the set of all elements <math>h \in G</math> such that conjugation by <math>h</math> commutes with conjugation by <math>g</math> for every <math>g \in G</math>. In other words, it is the subgroup comprising the elements whose induced inner automorphisms centralize all [[defining ingredient::inner automorphism]]s. | ||
# It is the second member of the [[defining ingredient::upper central series]] of <math>G</math>. | # It is the second member of the [[defining ingredient::upper central series]] of <math>G</math>. | ||
For more about the properties satisfied and not satisfied by this, see [[upper central series]]. | |||
Latest revision as of 18:24, 3 July 2013
This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions
Definition
Definition with symbols
The second center of a group , denoted , is defined in the following equivalent ways:
- It is the subgroup of such that contains the center of , and is the center of the quotient group .
- It is the set of all elements such that conjugation by commutes with conjugation by for every . In other words, it is the subgroup comprising the elements whose induced inner automorphisms centralize all inner automorphisms.
- It is the second member of the upper central series of .
For more about the properties satisfied and not satisfied by this, see upper central series.