| Summary: | Given a digital image (or digital object) <inline-formula><math display="inline"><semantics><mrow><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mo>,</mo><mi>X</mi><mo>⊂</mo><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mi>n</mi></msup></mrow></semantics></math></inline-formula>, this paper initially establishes a group structure of the set of self-<i>k</i>-isomorphisms of <inline-formula><math display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> with the function composition, denoted by <inline-formula><math display="inline"><semantics><mrow><mi>I</mi><mi>s</mi><msub><mi>o</mi><mi>k</mi></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> or <inline-formula><math display="inline"><semantics><mrow><mi>A</mi><mi>u</mi><msub><mi>t</mi><mi>k</mi></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>. In particular, let <inline-formula><math display="inline"><semantics><msubsup><mi>C</mi><mi>k</mi><mrow><mi>n</mi><mo>,</mo><mi>l</mi></mrow></msubsup></semantics></math></inline-formula> be a simple closed <i>k</i>-curve with <i>l</i> elements in <inline-formula><math display="inline"><semantics><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mi>n</mi></msup></semantics></math></inline-formula>. Then, the group <inline-formula><math display="inline"><semantics><mrow><mi>I</mi><mi>s</mi><msub><mi>o</mi><mi>k</mi></msub><mrow><mo stretchy="false">(</mo><msubsup><mi>C</mi><mi>k</mi><mrow><mi>n</mi><mo>,</mo><mi>l</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> is proved to be isomorphic to the standard dihedral group <inline-formula><math display="inline"><semantics><msub><mi>D</mi><mi>l</mi></msub></semantics></math></inline-formula> with order <i>l</i>. The calculation of this quantity <inline-formula><math display="inline"><semantics><mrow><mi>I</mi><mi>s</mi><msub><mi>o</mi><mi>k</mi></msub><mrow><mo stretchy="false">(</mo><msubsup><mi>C</mi><mi>k</mi><mrow><mi>n</mi><mo>,</mo><mi>l</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> is a key step for obtaining many new results. Indeed, it is essential for exploring many features of <inline-formula><math display="inline"><semantics><mrow><mi>I</mi><mi>s</mi><msub><mi>o</mi><mi>k</mi></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>. Furthermore, this quantity is proved to be a digital topological invariant. After proceeding with an <inline-formula><math display="inline"><semantics><mrow><mi>I</mi><mi>s</mi><msub><mi>o</mi><mi>k</mi></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>-action on <inline-formula><math display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, we investigate some properties of fixed point sets by this action. Finally, we explore various features of fixed point sets by this action from the viewpoint of digital <i>k</i>-curve theory. This paper only deals with <i>k</i>-connected digital images <inline-formula><math display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> whose cardinality is equal to or greater than 2. Besides, we discuss some errors that have appeared in the lilterature.
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