For a finite group $G$ and a prime number $p$, several normal subgroups are defined as follows:
- $O_p(G)$ = the largest normal $p$-subgroup of $G$ ($p$-core)
- $O^p(G)$ = the smallest normal subgroup of $G$ for which the quotient is a $p$-group ($p$-residual)
- $O_\infty(G)$ = the largest normal solvable subgroup of $G$ (solvable radical)
There are also $p'$ versions as well and more generally $\pi$ versions, but it doesn't matter here.
My question: why the symbol "$O$" is used in these?
Many symbols for constructing groups have obvious reasons such as $C_G(H)$ for a centralizer, $N_G(H)$ for a normalizer, $\Phi(G)$ for the Frattini subgroup, $D(G)$ for the derived subgroup, $Z(G)$ for the center (=Zentrum, a bit tricky) and so on and so forth. But I have no idea why the notation "$O$" is employed here. Do you know any reason? If there is no particular reason and it is just an accident, who started using it?