2020/12/03 by Churikov, Dmitry, Praeger, Cheryl E.
#05E18 #20B25 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2012.01773
For a positive integer k, a group G is said to be totally k-closed if in each of its faithful permutation representations, say on a set Ω, G is the largest subgroup of Sym(Ω) which leaves invariant each of the G-orbits in the induced action on Ω×…× Ω=Ωk. We prove that every abelian group G is totally (n(G)+1)-closed, but is not totally n(G)-closed, where n(G) is the number of invariant factors in the invariant factor decomposition of G. In particular, we prove that for each k≥2 and each prime p, there are infinitely many finite abelian p-groups which are totally k-closed but not totally (k-1)-closed. This result in the special case k=2 is due to Abdollahi and Arezoomand. We pose several open questions about total k-closure.