It is known that any compact connected Lie group with its left invariant framing is framed null-cobordant in the -component for any prime . In this paper we will prove that the 3-components of and are zero for , . Combining this with the previously known results on and consequently we see that any classical group has at most only the 2-component with some exceptions.
"On framed cobordism classes of classical Lie groups." J. Math. Soc. Japan 55 (4) 1033 - 1052, October, 2003. https://doi.org/10.2969/jmsj/1191418762