Aspects of Infinite Groups: A Festschrift in Honor of by Benjamin Fine, Gerhard Rosenberger, Dennis Spellman

By Benjamin Fine, Gerhard Rosenberger, Dennis Spellman

This e-book is a festschrift in honor of Professor Anthony Gaglione's 60th birthday. This quantity provides an exceptional mixture of learn and expository articles on a variety of elements of countless staff thought. The papers supply a wide assessment of current study in limitless crew concept more often than not, and combinatorial staff concept and non-Abelian group-based cryptography particularly. additionally they pinpoint the interactions among combinatorial workforce idea and mathematical common sense, in particular version concept.

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Let \7( G) be the subgroup of GI8IG generated by the set {g 181 gig E G}. The group \7(G) is a central subgroup of G 181 G (Brown and Loday4). The factor group GI8IGI\7(G) is called the nonabelian exterior square of G, denoted by GAG. For elements 9 and h in G, the coset (g 181 h)\7(G) is denoted 9 A h. Hence G 181 G is a central extension of GAG by \7 (G) and we have the short exact sequence 1 ----7 \7(G) a ----7 GAG ----7 1. The following theorem from Blyth, et al. 3 provides the basic structure for the nonabelian tensor square of a free nilpotent group of finite rank.

1 and provide formulas for ISn,c \ Sn,c-21 and ISn,c \ Sn,ll. 3). The nonabelian tensor square G I8i G of a group G was introduced by Brown and Loday4 following the ideas of Dennis5 and Miller 6 and is of topological significance. 1 to obtain a general structure description for this nonabelian tensor square. 1. Let G = Nn,c be the free nilpotent group of class c and rank n > 2. I). 2. 1 and then use a result of Gaglione and Spellman7 to find a formula for the number ISm,w \ Sm,11 of simple left normed commutators in Cm,w of weight at least 2.

Encoding is done within R so that elements of R represent messages. Bob wants to send the message r E R to Alice. He 36 randomly chooses an e E U(R) and sends Alice reo Alice randomly chooses f E U(R) and sends back fre. Bob knows (but presumably an attacker cannot figure out) e- l so he forms free- l = fr and sends this back to Alice. Alice applies f- l to get the message r. As a platform for this encryption method we propose the use of the ring of formal power series R« Xl, "',Xn » over a ring R in noncommuting variables Xl, ""xn' Although this can be done in an even more general context, for this study we will concentrate on rational formal power series, that is we consider the ring R to be the field of rational numbers Q.

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