# 2-idempotent 3-quasigroups with a conjugate invariant by Ji L.

By Ji L.

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**Extra resources for 2-idempotent 3-quasigroups with a conjugate invariant subgroup consisting of a single cycle of length four**

**Example text**

E. it is deleted). Of course, to be able to determine all the subgroups of a given group, it is necessary to use its complete graph as the basis for defining the subgraphs. Since in the group m g there are indirect isometries, this group does not give enantiomorphic modifications. , b) and "right" (d) form of an elementary asymmetric figure. For example, for the group 11, generated by a translation X, this results in the enantiomorphic friezes: bbbbbbbbbbbbbbbbbbbbbbbbbbbbb and dddddddddddddddddddddddddddddd.

1 Cayley diagrams 48 Theory of Isometric Symmetry Groups in E2 and Ornamental Art S R -o 6 D4(4m) *0 p-4 4 o 1/ o >o D 5 (5m) M? o o TO— o a 4 o »• D6(6m) Fig. 1 Rosettes Cayley diagrams and Ornamental Art The continuous symmetry group D ^ (com), the symmetry group of a circle, has a relative priority in ornamental art, both in the frequency of occurrence and in chronology. ), it is possible to find examples Symmetry Groups of Rosettes G20 49 of t h e oldest rosettes. Among them, the most i m p o r t a n t is the circle—a rosette with maximal symmetry.

T h a t Ri = R^1 while from the other relation follows: (RiP)2 = E=> R^R-L = P-1 => R^PRi = P'1 => PR = P"1. T h e glide reflection P is a transformation without invariant points, with an invariant line—the axis I of the glide reflection. j( / ). Since the axis of the glide reflection P^1 is the line — I , then from the previous relations we conclude t h a t R\( I ) = — / ; consequently, it follows t h a t the reflection line of the reflection JRI is perpendicular to the axis of the glide reflection P, and t h a t its axis I is non-polar (since there exists an indirect transformation, the reflection R\, which transforms it onto the oppositely oriented line — I ).