CA not implies nilpotent
This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., CA-group) need not satisfy the second group property (i.e., nilpotent group)
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It is possible to have a CA-group that is not a nilpotent group. Here, CA means that the centralizer of every non-identity element is abelian. Equivalently, it means that every proper subgroup is either a centerless group or an abelian group.
This also shows that:
The group symmetric group:S3, defined as the group of permutations on , is a CA-group that is nontrivial and centerless, and therefore not nilpotent. To see that it is CA, note that it is centerless, and all its proper subgroups are abelian.
Further information: infinite dihedral group
Consider the infinite dihedral group:
This is centerless, and therefore, not nilpotent. On the other hand, the centralizer of every non-identity element is abelian. To see this, note that: