| Summary: | Let <i>R</i> be a finite commutative chain ring with invariants <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>m</mi><mo>.</mo></mrow></semantics></math></inline-formula> The purpose of this article is to study j-diagrams for the one group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>=</mo><mn>1</mn><mo>+</mo><mi>J</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mo>(</mo><mi>R</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>π</mi><mo>)</mo></mrow></semantics></math></inline-formula> is Jacobson radical of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>.</mo></mrow></semantics></math></inline-formula> In particular, we prove the existence and uniqueness of j-diagrams for such one group. These j-diagrams help us to solve several problems related to chain rings such as the structure of their unit groups and a group of all symmetries of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msup><mi>π</mi><msup><mi>k</mi><mo>′</mo></msup></msup><mo>}</mo><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>k</mi><mo>′</mo></msup><mo>∣</mo><mi>k</mi></mrow></semantics></math></inline-formula>. The invariants <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>m</mi></mrow></semantics></math></inline-formula> and the Eisenstein polynomial by which <i>R</i> is constructed over its Galois subring determine fully the j-diagram for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>.</mo></mrow></semantics></math></inline-formula>
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