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<p>We show that the endomorphisms of a compact connected group that extend to endomorphisms of every compact overgroup are precisely the trivial one and the inner automorphisms; this is an analogue, for compact connected groups, of results due to Schupp and Pettet on discrete groups (plain or finite). A somewhat more surprising result is that if <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {A}\)</EquationSource> </InlineEquation> is compact connected and abelian, its endomorphisms extensible along morphisms into compact connected groups also include <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(-\textrm{id}\)</EquationSource> </InlineEquation> (in addition to the obvious trivial endomorphism and the identity). Connectedness cannot be dropped on either side in this last statement.</p>

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Extensible Endomorphisms of Compact Groups

  • 期刊论文

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  • 发表日期：

    2026年3月2日

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  • Alexandru Chirvasitu

摘要

We show that the endomorphisms of a compact connected group that extend to endomorphisms of every compact overgroup are precisely the trivial one and the inner automorphisms; this is an analogue, for compact connected groups, of results due to Schupp and Pettet on discrete groups (plain or finite). A somewhat more surprising result is that if \(\mathbb {A}\) is compact connected and abelian, its endomorphisms extensible along morphisms into compact connected groups also include \(-\textrm{id}\) (in addition to the obvious trivial endomorphism and the identity). Connectedness cannot be dropped on either side in this last statement.