Center not is surjective endomorphism-balanced

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This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) does not always satisfy a particular subgroup property (i.e., surjective endomorphism-balanced subgroup)
View subgroup property satisfactions for subgroup-defining functions | View subgroup property dissatisfactions for subgroup-defining functions

Statement

Statement with symbols

It is possible to have a group G and a surjective endomorphism \sigma of G such that the restriction of \sigma to the center Z(G) is not surjective as an endomorphism of Z(G).

Related facts

Other proofs of similar subgroup-defining functions not being surjective endomorphism-balanced

Proofs of similar subgroup-defining functions being strictly characteristic

Facts used

  1. Isomorphic to inner automorphism group not implies centerless: We can have a group G such that the center Z(G) is nontrivial, but there is an isomorphism between G/Z(G) and G.

Proof

Proof using fact (1)

Let G be an example group for fact (1). Let \alpha:G \to G/Z(G) and the quotient map and \varphi:G/Z(G) \to G be an isomorphism. Then, \sigma = \varphi \circ \alpha is a surjective endomorphism of G. On the other hand, the restriction of \sigma to Z(G) is the trivial map, which is not surjective by the assumption that Z(G) is a nontrivial group.