| Summary: | We find that the degeneracies and many peculiarities of the DNA genetic code may be described thanks to two closely related (fivefold symmetric) finite groups. The first group has signature <inline-formula><math display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><msub><mi mathvariant="double-struck">Z</mi><mn>5</mn></msub><mo>⋊</mo><mi>H</mi></mrow></semantics></math></inline-formula> where <inline-formula><math display="inline"><semantics><mrow><mi>H</mi><mo>=</mo><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub><mo>.</mo><msub><mi>S</mi><mn>4</mn></msub><mo>≅</mo><mn>2</mn><mi>O</mi></mrow></semantics></math></inline-formula> is isomorphic to the binary octahedral group <inline-formula><math display="inline"><semantics><mrow><mn>2</mn><mi>O</mi></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>S</mi><mn>4</mn></msub></semantics></math></inline-formula> is the symmetric group on four letters/bases. The second group has signature <inline-formula><math display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><msub><mi mathvariant="double-struck">Z</mi><mn>5</mn></msub><mo>⋊</mo><mi>G</mi><mi>L</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and points out a threefold symmetry of base pairings. For those groups, the representations for the 22 conjugacy classes of <i>G</i> are in one-to-one correspondence with the multiplets encoding the proteinogenic amino acids. Additionally, most of the 22 characters of <i>G</i> attached to those representations are informationally complete. The biological meaning of these coincidences is discussed.
|
|---|