Interdependencies of group properties (awaiting review)

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.

Since cyclic groups are always abelian, Z-groups are by definition A-groups, and similarly metacyclic groups are metabelian as well.

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 since all their subgroups (including their Sylow subgroups) are abelian.

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

Supersolvable groups are monomial (see Corollary 3.5 in Chapter 2 of [MR:0655785]).

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

Since the definition of a metabelian group ensures the existence of a normal subgroup $N \lhd G$ so that $G/N$ is abelian, metabelian groups also satisfy the definition of solvable via the chain of subgroups $\langle e\rangle \leq N \leq G$.

If all the Sylow subgroups are cyclic, then the group is metacyclic. Hence Z-groups are metacyclic (see Theorem 9.4.3 in [MR:0103215]).

Nonabelian 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, nonabelian simple groups are also almost simple and quasi-simple.