Markku Niemenmaa

On multiplication groups of loops

The concept of multiplication groups of quasigroups was introduced by Albert Cl] and the connection between quasigroups and corresponding multiplication groups has been studied by Bruck [6], Smith [20] and Ihringer [14, 151. While studying the multiplication group of a loop Q (a quasigroup with neutral element) a central role is played by the stabilizer of the neutral element. This subgroup 1(Q) of the multiplication group is called the inner mapping group of Q. If Q is a group then it is clear that r(Q) consists of the inner automorphisms of Q. We also know that a loop Q is an abelian group if and only if 1(Q) = I. In this paper we study some properties of the inner mapping group and we also give a partial answer to the question: What are the multiplication groups of loops? This question is closely connected to certain transversal conditions. Sections 2 and 3 are devoted to investigating these conditions and in Section 4 we characterize multiplication groups of loops with the aid of these conditions. In the same section we prove one of our main results: If Q is a finite loop whose inner mapping group is cyclic, then Q is an abelian group. Finally, in Section 5 we use the properties of the inner mapping group in order to show that certain groups are not multiplication groups of loops. We also give examples of groups which are multiplication groups of loops. Our notation is standard and for basic facts about groups and loops WC refer to [4,7, 133. 112

FINITE LOOPS WITH NILPOTENT INNER MAPPING GROUPS ARE CENTRALLY NILPOTENT

In this article we show that finite loops with nilpotent inner mapping groups are centrally nilpotent.

ON LOOPS WHICH HAVE DIHEDRAL 2-GROUPS AS INNER MAPPING GROUPS

In this paper we consider the situation that a group G has a subgroup H which is a dihedral 2-group and with connected transversals A and B in G. We show that G is then solvable and moreover, if G is generated by the set A U B, then H is subnormal in G. We apply these results to loop theory and it follows that if the inner mapping group of a loop Q is a dihedral 2-group then Q is centrally nilpotent.

On connected transversals to subgroups whose order is a product of two primes

We consider finite groups which have connected transversals to subgroups whose order is a product of two primes p and q. We investigate those values of p and q for which the group is soluble. We can show that the solubility of the group follows if qD 2 and p 61, qD 3 and p 31, qD 5 and p 11. We then apply our results on loop theory and we show that if the inner mapping group of a finite loop has order pq where p and q are as above then the loop is soluble. c 1997 Academic Press Limited

On Connected Transversals to Nonabelian Subgroups

In this paper we consider groups G which have connected transversals to nonabelian subgroups whose order is a product of two odd primes p and q, where pq andp= 2 qm+ 1. In our main theorem we show thatG is then solvable. We apply our results to loop theory and it follows that if the inner mapping group of a finite loop has order pq, where p and q are as previously given, then the loop is solvable.

On conjugacy classes in finite loops

The role of the conjugacy relation is certainly important in the structure theory of groups. Here we study this relation in a considerably more general setting, namely in the theory of loops. We first recall some basic facts about quasigroups, their multiplication groups, their inner mapping groups and the conjugacy relation. After this we estimate the size and the number of the conjugacy classes and we study the structure of loops having only two conjugacy classes. Finally, the values of the centraliser function are discussed.

Length functions and fuzzy groups

Fuzzy sets were introduced by Zadeh [7] in 1965 and fuzzy groups by Rosenfeld [4] in 1971. Fuzzy groups were also considered by Anthony and Sherwood [I] in 1979 and in 1981 the theory was developed by Das [2]. As in the paper of Das [2] we also show how a fuzzy subgroup of a group G is determined by a chain of subgroups of G. However, we wish to point out that fuzzy subgroups are constructed in a way which is analogous to that of constructing certain length functions of G (Wilkens [S, 61). Thus in Section 3 we consider the theory of length functions and we show the connection between certain length functions and fuzzy subgroups of a group. In Section 4 we apply the structural results of Section 3 on finite solvable groups and we gain a generalization of a theorem of Das [2]. Our notation is standard. However, we use the symbol e to denote the identity element of a group G and I denotes a real number. We start with a preliminary section where we briefly analyse the definition of a fuzzy subgroup given by Rosenfeld [4] (naturally we use this definition of a fuzzy subgroup throughout the paper).

On some properties of a check digit system

In this paper, we consider check digit systems which are based on the use of elementary abelian p-groups of order pk. The work is inspired by a recently introduced check digit system for hexadecimal numbers. By interpreting its check equation in terminology of matrix algebra, we generalize the idea to build systems over a group of order pk, while keeping the ability to detect all the 1) single errors, 2) adjacent transpositions, 3) twin errors, 4) jump transpositions and 5) jump twin errors. Besides, we consider two categories of jump errors: t-jump transpositions and t-jump twin errors, which include and further extend the double error types of 2)-5). In particular, we explore the capacity range of the system to detect these two kinds of generalized jump errors, and demonstrate that it is 2k - 3 for p = 2 and (pk -1)/2-2 for an odd prime p. Also, we show how to build such a system that detects all the single errors and these two kinds of double jump-errors within the capacity range.

A check digit system for hexadecimal numbers

We present a new check digit system for hexadecimal numbers which is based on the use of a suitable automorphism of the elementary abelian group of order sixteen. Our system is able to detect all single errors, adjacent transpositions, twin errors, jump transpositions and jump twin errors.

On the Error Detection Capability of One Check Digit

In this paper, we study a check digit system which is based on the use of elementary abelian p-groups of order pk. This paper is inspired by a recently introduced check digit system for hexadecimal numbers. By interpreting its check equation in terminology of matrix algebra, we generalize the idea to build systems over a group of order pk, while keeping the ability to detect all the: 1) single errors; 2) adjacent transpositions; 3) twin errors; 4) jump transpositions; and 5) jump twin errors. Besides, we consider two categories of jump errors: 1) t-jump transpositions and 2) t-jump twin errors, which include and further extend the double error types of 2)-5). In particular, we explore Rc, the maximum detection radius of the system on detecting these two kinds of generalized jump errors, and show that it is 2k-2 for p=2 and (pk-1)/2-1 for an odd prime p. Also, we show how to build such a system that detects all the single errors and these two kinds of double jump-errors within Rc.