On the Efficiency of Wreath Products of Groups

On defining groups efficiently without using inverses

“…In [2,3,4], efficient semigroup presentations for various groups are found. The groups considered include the finite abelian groups, dihedral groups, (projective) special linear groups and wreath products.…”

“…The groups considered include the finite abelian groups, dihedral groups, (projective) special linear groups and wreath products. An efficient group presentation for C n C m with n odd and m even is [3] for details. Here we illustrate constructions described in Propositions 2·4 and 2·5 by giving two new examples: the wreath products C n C m (n odd, m even) of two cyclic groups and the projective special unitary group P SU (3,3) Example 3·1.…”

“…In fact, our Main Theorem implies that a finite group is efficient as a group if and only if it is efficient as a semigroup. Here we illustrate constructions described in Propositions 2•4 and 2•5 by giving two new examples: the wreath products C n C m (n odd, m even) of two cyclic groups and the projective special unitary group P SU (3,3) Example 3•1. An efficient group presentation for C n C m with n odd and m even is…”

“…2) ; see [3] for details. Observe that this presentation satisfies the conditions in Propostion 2•4; more precisely the relation a n 1 a m 2 = 1 contains no inverses, contains both generators and contains the square of a generator.…”

The efficiency of standard wreath product

“…Remark 1.4. After this paper was submitted it was brought to our attention that a special case of our theorem was obtained independently in [2].…”