]> 2026-08-26T03:13:07+00:00 Group acts as automorphisms by conjugation 0 en Group acts as automorphisms by conjugation Characteristic implies normal 0 en Characteristic implies normal Cyclic automorphism group implies abelian 0 en Cyclic automorphism group implies abelian Normality is strongly join-closed 0 en Normality is strongly join-closed N-abelian implies every nth power and (n-1)th power commute 0 en N-abelian implies every nth power and (n-1)th power commute Nilpotent automorphism group implies nilpotent of class at most one more 0 en Nilpotent automorphism group implies nilpotent of class at most one more Solvable automorphism group implies solvable of derived length at most one more 0 en Solvable automorphism group implies solvable of derived length at most one more Abelian automorphism group implies class two 0 en Abelian automorphism group implies class two Center is local powering-invariant 0 en Center is local powering-invariant Uses 132 en Uses