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

A characteristic subgroup is automatically normal, since it is stabilized by all automorphisms including the inner automorphisms.

A cyclic group is always abelian by the way the group operation for cyclic groups is defined. An abelian group is always nilpotent since the group is its own center and thus an upper central series for the group exists. Nilpotent groups are are solvable because the upper central series defining a nilpotent group has abelian subquotients.

A stem subgroup is central by definition, and any central subgroup is abelian since its elements commute with the whole group.

A Sylow subgroup is a $p$-group and thus nilpotent.

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- Last edited by David Roe on 2021-09-30 17:25:06

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