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.
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> 
> 


Thank you very much.



Searching for theorems characterizing when $O_p(G)$ is trivial / non-trivial
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.


