## K. W. Johnson, M. Munywoki, Jonathan D. H. Smith

*The upper triangular algebra loop of degree $4$*

Comment.Math.Univ.Carolin. 55,4 (2014) 457-470.**Abstract:**A natural loop structure is defined on the set $U_4$ of unimodular upper-triangular matrices over a given field. Inner mappings of the loop are computed. It is shown that the loop is non-associative and nilpotent, of class 3. A detailed listing of the loop conjugacy classes is presented. In particular, one of the loop conjugacy classes is shown to be properly contained in a superclass of the corresponding algebra group.

**Keywords:** algebra group; quasigroup; loop; supercharacter; fusion scheme

**AMS Subject Classification:** 20N05

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