group. Then $G$ is acceptable (or strongly acceptable) if and only if its derived subgroup $G_{der}$ is.

Now we show Proposition \[P:centralizer2\].

We first give the proof for the strongly acceptability. Suppose $G$ is strongly acceptable. Write $H=
Z_{G}(A)$. Let $\Gamma$ be a group, and $\phi_{1},\phi_{2}:\Gamma\rightarrow H$ be two element-conjugate homomorphisms. Put $\Gamma'=\Gamma\times A$. Define $\phi'_{i}: \Gamma'\rightarrow G$ ($i=1,2$) by $$\phi'_{i}(\gamma,a)=\phi_{i}(\gamma)a,\ \forall\gamma\in\Gamma,a\in A.$$ Apparently, $\phi'_{1}$ and $\phi'_{2}$ are element-conjugate homomorphisms. Since $G$ is strongly acceptable, there exists $g\in G$ such that $$\phi'_{2}(\gamma,a)=\Ad(g)(\phi'_{1}(\gamma,a)),\ \forall(\gamma,a)\in\Gamma'.$$ Applying to $\gamma=1$ and $a\in A$, we get $g\in Z_{G}(A)=H$. Applying to $a=1$ and $\gamma\in\Gamma$, we get $\phi_{2}(\gamma)=\Ad(g)(\phi_{1}(\gamma))$ for any $\gamma\in\Gamma$. The just means, $\phi_1$ and $\phi_2$ are globally conjugate. Thus, $H$ is strongly acceptable.

For the acceptability. We could take $m>>1$ such that $Z_{G}(A(m))=Z_{G}(A)$, where $$A(m)=\{x\in A:
x^{m}=1\}.$$ Using $A(m)$ instead of $A$ in the above argument, the proof is the same.

Examples of “SCF" subgroups
===========================

In [@Griess], Griess showed several “SCF” pairs. The following proposition follows from results shown in [@Griess].

\[T:Griess\] The following pairs are “SCF” pairs.

-   $\O(n)\subset\U(n)$ and $\Sp(n)\subset\U(2n)$ (cf. [@Griess Theorem 2.3]);

-   $\G_2\subset\O(7)$ and $\G_2\subset\Pin(7)$ (cf. [@Griess Theorem 1.1]);

-   $\F_4\subset\E_6$ (cf. [@Griess Theorem 1.5]).

In this section, we 