# Questions tagged [derived-subgroup]

Derived subgroup (or commutator subgroup) of a group is the subgroup generated by all commutators of that group (thus being the verbal subgroup for the word $[a, b]$). To be used with the tag [group-theory]

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### Can a group have a cyclical derived series?

Note: this question was asked on Math Overflow, and was subsequently answered in the affirmative. Given any group $G$, one can consider its derived series G = G^{(0)}\rhd G^{(1)}\rhd G^{(2)}\rhd\...
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### Does there exist a finite group that is both perfect and immaculate?

A group $G$ is called perfect iff $G’ = G$. A finite group $G$ is called immaculate iff its order is equal to the sum of orders of its proper normal subgroups. Does there exist a finite group $G$, ...
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### Does there exist a non-trivial group that is both perfect and complete?

A group $G$ is called perfect iff $G’ = G$. A group $G$ is called complete iff $Z(G) = \{e\}$ and $Aut(G) \cong G$. Does there exist a non-trivial group $G$, that is both perfect and complete at the ...