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Certain properties of groups automatically imply other properties. We briefly explain the connections here.

A cyclic group is always abelian by the way the group operation for cyclic groups is defined. Cyclic groups are also automatically Z-groups since all their subgroups (including their Sylow subgroups) are cyclic.

An abelian group is always nilpotent since the group is its own center and thus an upper central series for the group exists. In addition, abelian groups are A-groups s since all their subgroups (including their Sylow subgroups) are abelian.

Nilpotent groups which are also finite (as is the case for groups in this database) are supersolvable because the upper central series defining a nilpotent group consists of abelian groups which, in the case of finite groups, may be expanded to a chain with cyclic factors.

Supersolvable groups which are also finite (as is the case for groups in this database) are monomial (reference needed).

Monomial groups are also solvable by a result of Taketa [MR:1568284 ].

Non-abelian simple groups are always perfect groups since the commutator subgroup of a group is a normal subgroup and hence must be the whole group in this case. In addition, such groups are also almost simple and quasi-simple.

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Knowl status:
• Review status: beta
• Last edited by Jennifer Paulhus on 2021-07-16 14:38:17
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Not referenced anywhere at the moment.

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