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The group {1, −1} above and the cyclic group of order 3 under ordinary multiplication are both examples of abelian groups, and inspection of the symmetry of their Cayley tables verifies this. In contrast, the smallest non-abelian group, the dihedral group of order 6, does not have a symmetric Cayley table.
Thus, normalizing a Cayley table (putting the border headings in some fixed predetermined order by permuting rows and columns including the headings) preserves the isotopy class of the associated Latin square. Furthermore, if two normalized Cayley tables represent isomorphic quasigroups then their associated Latin squares are also isomorphic.
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2 Cayley table. 3 Properties. 4 Matrix representations. ... Free group; Modular groups. PSL ... A generalized quaternion group Q 4n of order 4n is defined by the ...
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Cayley table of the alternating group A 4 as a subgroup of S 4. See: Subgroups of S 4. This group shows even permutations of 4 elements - or rotations of the tetrahedron respectively. (see Tetrahedral symmetry)
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
Since the free group has no relations, the Cayley graph has no cycles: it is the 4-regular infinite tree. It is a key ingredient in the proof of the Banach–Tarski paradox. More generally, the Bethe lattice or Cayley tree is the Cayley graph of the free group on generators.