Complete solvable group
This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: complete group and solvable group
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A complete solvable group is a group that satisfies both these conditions:
- It is a complete group, i.e., it is both a centerless group and a group in which every automorphism is inner.
- It is a solvable group.
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|solvable group in which every automorphism is inner||we no longer require the group to be centerless||there exist nilpotent nontrivial groups, such as cyclic group:Z2, in which every automorphism is inner|||FULL LIST, MORE INFO|
|complete group||there are complete non-solvable groups such as symmetric group:S5||complete group|complete|
|solvable group||Group whose automorphism group is solvable, Solvable group in which every automorphism is inner|FULL LIST, MORE INFO|
|group in which every automorphism is inner||cyclic group:Z2||Solvable group in which every automorphism is inner|FULL LIST, MORE INFO|
|group whose automorphism group is solvable|||FULL LIST, MORE INFO|