View Full Document

# Study References

View Full Document
View Full Document

5 views

Unformatted text preview:

Comment Math Univ Carolin 41 2 2000 245 249 A loops close to code loops are groups Ales Dra pal Abstract Let Q be a diassociative A loop which is centrally nilpotent of class 2 and which is not a group Then the factor over the centre cannot be an elementary abelian 2 group Keywords A loop central nilpotency Osborn problem Classification 20N05 This modest note concerns diassociative A loops that are centrally nilpotent of class 2 Its basic result can be expressed in the following way Proposition 1 Suppose that Q is a diassociative A loop with a central subloop N If Q N is a group of exponent 2 then Q is a group The result is an offshoot of my interest in code loops and I hope that it will help to stimulate further research in the indicated direction It can be also understood in the context of the Osborn problem Decide if every finite diassociative A loop is a Moufang loop Osborn 6 solved this problem affirmatively for commutative diassociative A loops and some progress in the general case has been recently reported by J D Phillips 7 A subloop N of a loop Q is central if all its elements associate and commute with all elements of Q Suppose that Q N V and that N and V are fixed The group N is abelian by definition and we shall assume that V is an abelian group as well though some of our initial observations can be easily generalized to the case of non abelian groups The loop Q is obviously isomorphic to one of the loops Q where V V N is a mapping with u 0 0 u 0 for all u V and where the binary operation of Q is defined by a u b v a b u v u v for all a b N and u v V A loop Q is said to be a code loop 5 4 1 if N 2 and V is an elementary abelian 2 group Statements about code loops are often proved by computations in Q and this approach will be used also here It is clear that one could Research partially supported by Grant Agency of the Czech Republic grant number 201 99 0263 and by an institutional grant of the Czech Republic code CEZ J13 98 113200007 245 246 A Dra pal

## Access the best Study Guides, Lecture Notes and Practice Exams

Unlocking...
Sign Up

Join to view Study References and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?