On some algebras of n-generated quaternion groups
by
Kalcho Todorov
South-West University Bulgaria

By B. H. Neuman [2] and W. Magnus and al. [1 ] the quaternion group Q was be considered as a two-generated group - Q  =  <i,  j | i  =  jij,   j  = iji>.
By generalization the latter presentation of the quaternion group we get the groups Q_n  = <a₁,  a₂,   ...  |  a_i  =  a_ja_ia_j,   i  =/=  j,  i, j  =  1,  2,   ... >, containing the quaternion group Q  =  Q₂ as subgroup and having a number of its basic properties (see Todorov [3]). In particular, the groups Q_n, their algebras H_n (over the field R of all real numbers) and the integral elements of the algebras H_n can be applied in the theory of Hadamard matrices (see Todorov [4]).
The following question is based on Frobenius theorem applied on quaternion algebras H_n, for n > 2: How to decompose H_n algebras as a sum of undecomposeble real, complex and quaternion algebras. The properties of the quaternion group Q_n show that basic tools in research of the answer are H_n when n is 2, 3, 4 and 5. In [5] the author gives full description of H₃. The present paper contains description of zero dividers of H₄ and H₅.

1. Magnus, W., A. Karras, D. Solitar, Combinatorical Group Theory, Interscience New York, 1966
2.  Neumann, B. H., Some Remarks on Semigroup Presentations. Can. J. Math. 19(1967), 1018-1026
3.  Todorov, K., Generalised Quaternion Group. MTA SZTAKI Közlemények 38(1988), 53-65
4.  Todorov, K., On the Generalised Quaternion Group and the Hadamard Matrices MTA SZTAKI Közlemények, 38(1988), 67-79
5. Todorov, K., On a generalization of the quaternion group and of the quaternion algebra. Potsdamer Forschungen. Reihe B. Heft 62, 1989, 101 - 107

Date received: May 31, 2000