Projects per year
Abstract
A finite group G is coprimely invariably generated if there exists a set of generators {g_{1},. .,g_{u}} of G with the property that the orders g_{1},. .,g_{u} are pairwise coprime and that for all x_{1},. .,x_{u}∈G the set {g_{1}^{x1},. .,g_{u}^{xu}} generates G.
We show that if G is coprimely invariably generated, then G can be generated with three elements, or two if G is soluble, and that G has zero presentation rank. As a corollary, we show that if G is any finite group such that no proper subgroup has the same exponent as G, then G has zero presentation rank. Furthermore, we show that every finite simple group is coprimely invariably generated by two elements, except for O_{8}^{+}(2) which requires three elements.
Along the way, we show that for each finite simple group S, and for each partition π_{1},. .,π_{u} of the primes dividing S, the product of the number k_{πi}(S) of conjugacy classes of πielements satisfies. ∏_{i=1}u k_{πi}(S)≤S/2OutS.
We show that if G is coprimely invariably generated, then G can be generated with three elements, or two if G is soluble, and that G has zero presentation rank. As a corollary, we show that if G is any finite group such that no proper subgroup has the same exponent as G, then G has zero presentation rank. Furthermore, we show that every finite simple group is coprimely invariably generated by two elements, except for O_{8}^{+}(2) which requires three elements.
Along the way, we show that for each finite simple group S, and for each partition π_{1},. .,π_{u} of the primes dividing S, the product of the number k_{πi}(S) of conjugacy classes of πielements satisfies. ∏_{i=1}u k_{πi}(S)≤S/2OutS.
Original language  English 

Pages (fromto)  34533465 
Number of pages  13 
Journal  Journal of Pure and Applied Algebra 
Volume  219 
Issue number  8 
Early online date  12 Dec 2014 
DOIs  
Publication status  Published  Aug 2015 
Fingerprint
Dive into the research topics of 'Coprime invariable generation and minimalexponent groups'. Together they form a unique fingerprint.Projects
 1 Finished

Solving word problems: Solving word problems via generalisations of small cancellation
RoneyDougal, C. & Neunhoeffer, M.
1/10/11 → 30/09/14
Project: Standard