Abelian-quotient not implies cocentral

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., abelian-quotient subgroup) need not satisfy the second subgroup property (i.e., cocentral subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about abelian-quotient subgroup|Get more facts about cocentral subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property abelian-quotient subgroup but not cocentral subgroup|View examples of subgroups satisfying property abelian-quotient subgroup and cocentral subgroup

Statement

It is possible to have a group G and a normal subgroup H such that G/H is an abelian group, so H is an abelian-quotient subgroup, but HZ(G) is not equal to G, i.e., H is not a cocentral subgroup of G.