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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.
A n: the alternating group of degree n, containing the even permutations of n elements, of order 1 for n = 0, 1, and order n!/2 otherwise; Dic n or Q 4n: the dicyclic group of order 4n. Q 8: the quaternion group of order 8, also Dic 2; The notations Z n and Dih n have the advantage that point groups in three dimensions C n and D n do not have ...
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2 Cayley table. 3 Properties. 4 Matrix representations. 5 Galois group. ... A generalized quaternion group Q 4n of order 4n is defined by the presentation [3]
V is the symmetry group of this cross: flipping it horizontally (a) or vertically (b) or both (ab) leaves it unchanged.A quarter-turn changes it. In two dimensions, the Klein four-group is the symmetry group of a rhombus and of rectangles that are not squares, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180° rotation.
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A 4 is isomorphic to PSL 2 (3) [1] and the symmetry group of chiral tetrahedral symmetry. A 5 is isomorphic to PSL 2 (4), PSL 2 (5), and the symmetry group of chiral icosahedral symmetry. (See [1] for an indirect isomorphism of PSL 2 (F 5) → A 5 using a classification of simple groups of order 60, and here for a direct proof). A 6 is ...