Every reasonably sized matrix group has an injective homomorphism into the group of all bijections of the natural numbers. However, not every reasonably sized simple group has an injective homomorphism into .
@article{bwmeta1.element.bwnjournal-article-fmv164i1p35bwm, author = {Robert Kallman}, title = {Every reasonably sized matrix group is a subgroup of S $\infty$}, journal = {Fundamenta Mathematicae}, volume = {163}, year = {2000}, pages = {35-40}, zbl = {0967.20002}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv164i1p35bwm} }
Kallman, Robert. Every reasonably sized matrix group is a subgroup of S ∞. Fundamenta Mathematicae, Tome 163 (2000) pp. 35-40. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv164i1p35bwm/
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