Center not is surjective endomorphism-balanced

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)
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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

• Baer norm not is surjective endomorphism-balanced
• Wielandt subgroup not is surjective endomorphism-balanced

Proofs of similar subgroup-defining functions being strictly characteristic

• Center is strictly characteristic
• Baer norm is strictly characteristic
• Wielandt subgroup is 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.