Let $G$ be a finite group. Let $p$ be a prime.
Let $O_p(G)$ be the $p$-core of $G$.
Are there any theorems known saying something like
$O_p(G)$ is trivial, if and only if ... and
$O_p(G)$ is non-trivial, if and only if ..., respectively ?
I am especially interested in the case $p=2$.
If not, criteria ensuring that $O_p(G)$ is trivial / non-trivial would also be interesting.
Thank you very much.
There are many such theorems. By the way, I would say that your definition of $p$-core may be non-standard if you are using it to denote the largest normal $p$-subgroup of $G$. I think many people would use $p$-core of $G$ to be $O_{p^{\prime}}(G)$, the largest normal subgroup of $G$ of order co prime to $p$. Certainly when $p = 2$, the largest normal subgroup of $G$ of order prime to $2$ is denoted by $O(G)$ and called the core of $G.$
Anyway, assuming you really do mean the largest normal $p$-subgroup of $G$ when you write $O_{p}(G)$, we have: $O_{p}(G)$ is the intersection of all Sylow $p$-subgroups of $G$. If $G$ has an Abelian Sylow $p$-subgroup $P$ then $O_{p}(G) \neq 1$ if and only if $P \cap P^{g} > 1$ for all $g \in G$. This is a theorem of J. Brodkey, and it does not work in general for non-Abelian Sylow $p$-subgroups. A famous theorem of R. Baer and M Suzuki is that if there is a non-idenity element $x \in G$ such that $\langle x,x^{g} \rangle $ is a $p$-group for all $g \in G$, then $x \in O_{p}(G)$, so that $O_{p}(G) \neq 1.$ When $p =2$, a consequence of this when $p =2$ is that if $t$ is an involution (element of order $2$) which does not invert any non-identity element of odd order, then $t \in O_{2}(G)$.
I could go on, but reading a graduate level group theory text should give you many more examples.
Theorem 2 in my preprint "Group Orders That Imply Existence of Nontrivial Normal $p$-Subgroups" shows that if $|G| = p^s m$ and $p \nmid \Gamma(m)$ (defined therein), then either $O_p (G) \ne 1$, or $G$ is not solvable.
