We show that 𝔤 ≤ c(Sym(ω)) where 𝔤 is the groupwise density number and c(Sym(ω)) is the cofinality of the infinite symmetric group. This solves (the second half of) a problem addressed by Thomas.
"The cofinality of the infinite symmetric group and groupwise density." J. Symbolic Logic 68 (4) 1354 - 1361, December 2003. https://doi.org/10.2178/jsl/1067620191