Certain properties of subgroups 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. 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.
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- Last edited by David Roe on 2021-09-30 17:25:06