By Mazurov V. D.

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74], [13, Ch. XI], [125, Ch. II], [165, Th. 5]. 5 The construction of a unipotent loop from an idempotent quasigroup goes back to Bruck [19]. For related constructions and further discussion, see [28, Ex. 5]. 6 Readers unfamiliar with elementary geometric concepts are referred to [62]. 2] helps elucidate why Steiner’s name is attached to the triple systems. Note that Fig. 3 in [62] only shows 10 of the 12 blocks. 7 Zorn’s vector-matrix algebra was presented in [179]. For more details on the octonions, see [33] and [50].

Since the permutation group G is transitive, the stabilizers of elements of Q are all conjugate to each other. 1. Thus Ge in this case is the inner automorphism group Inn Q of Q. If Q is a pique with pointed idempotent e, the stabilizer Ge of the pointed idempotent is called the inner multiplication group (or inner mapping group) Inn Q of Q. As the following example shows, Inn Q need not consist entirely of automorphisms of Q, even if e is the identity element of a loop Q. 3. It has R(1) = (01)(243), R(2) = (02)(134), and R(3) = (03)(142), whence R(1)R(2)R(3) = (24) ∈ Inn Q.

Define 0η = 000 and 1η = 111. Thus C = {000, 111}. 31) by δ −1 {000} = {000, 001, 010, 100} and δ −1 {111} = {111, 110, 101, 011} (“majority vote”). Provided that at most one letter of the emitted codeword gets © 2007 by Taylor & Francis Group, LLC MULTIPLICATION GROUPS 47 corrupted in the channel, the decoder is able to recover the codeword. One may extend this scheme to channels of greater odd length. For further analysis, it is convenient to put an abelian group structure (A, +, 0) on the alphabet A.