Cayley Tables for Small Groups and Moufang Loops
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A group is a set of elements with a closed, invertible, and associative binary operation with a unit element. The associative property is for all in the group. A group is commutative (or Abelian) if for all in the group. (Often then the operation is written .)[more]
A loop is a set of elements with a closed, invertible binary operation with a unit. In other words, a loop is a group provided it is associative. A Moufang loop is a loop with a weak form of associativity, for all in the Moufang loop. So all groups are Moufang loops; Moufang loops are used to define vector-division tables (usually with divisors of zero). All Moufang loops are alternative (or square associative) , but some alternative loops lack the Moufang property.
Loops can be expressed as Cayley index tables with the first row and column containing the indices in that order and with each index occurring once in each row and column. They can also be expressed as symbolic tables using letters. This Demonstration shows both forms, with colored or plain backgrounds (the colors might be hard to read for large ).
When has several factors, there may be many groups; reordering the indices (or elements) gives many isomorphs representing the same group. I choose a "protoloop" (my name for a preferred isomorph, chosen to display structure) to represent each group as a standard table. I provide an unambiguous group name of the form gxxyy (e.g., g1607), where yy is the index in the list of groups of length xx in the GAP atlas. (The GAP atlas lists all groups up to a certain size.) I include a few examples that are not groups after the groups using Mxxyy for the Moufang loops, nxxyy for general systems, and Altxxyy for alternative loops that are not Moufang loops.
In most cases, the table is created by an "incantation" (my name for any formula that creates a loop table) involving the following procedures: (1) ca[m, a : 1] creates a loop of length ; gives the cyclic group ; gives a dihedral group ; gives a generalized quaternion group . Other values may give other groups. (2) co[g, h, a : 1] "composes" the groups and to get ; if , the result may be an indirect composition or . (3) cadi and md create loops via generalized Cayley-Dickson and Moufang "doubling".
I do not know incantations in this form for a few of the groups in this Demonstration, so some are supplied as named tables. 15 (out of 51) 32-element groups, and two (out of 14) 36-element groups are omitted. A few (nonassociative) Moufang loops may be missing. The Demonstration is based on "Groups, Loops, & Hoop Algebras" (see Related Link), which contains many more groups, loops, and incantations.
Select a table length and an index. The table will appear. Use "show" to choose between different displays. The group name gxxyy and a mnemonic (usually showing the group type or structure) appear above the table. The incantation appears below it, together with an identifier. The identifier is a "generalized table signature" unique to isomorphs of most groups.[less]
Contributed by: Roger Beresford (March 2011)
Open content licensed under CC BY-NC-SA
Two routines implement extended versions of Cayley-Dickson doubling, (e.g., tripling the quaternion algebra with gives Alt1207), and Moufang doubling, (which generates a number of nonassociative Moufang loops by doubling non-Abelian groups, e.g., creating M1206 from ).
See also the Wikipedia article "Cayley–Dickson Construction" and the GAP home page "GAP—Groups, Algorithms, Programming—a System for Computational Discrete Algebra".