Let $G$ be a finite group, and fix a prime $p$. For conjugate elements $a,b\in G$, define: $$d_c(a,b)=\min\{\nu_p(|H|)\mid H\triangleleft G,\text{ there is }h\in H\text{ such that }a^h=b\}$$ where, as usual, $\nu_p(n)$ is the exponent of the largest power of $p$ dividing $n$. "$H\triangleleft G$" means "$H$ is a normal subgroup of $G$".
If $a$ and $b$ are non-conjugate, define $d_c(a,b)=\infty$.
Then:
- $d_c(a,a)=0$. If $\mathbf{O}_{p'}(G)=1$, then $d_c(a,b)=0$ if and only if $a=b$. ($\mathbf{O}_{p'}(G)$ is the unique largest normal subgroup of $G$ whose order is not divisible by $p$.)
- $d_c(a,b)=d_c(b,a)$
- $d_c(a,c)\leq d_c(a,b)+d_c(b,c)$.
- $d_c(a^x, b^x)=d_c(a,b)$
Proof of triangle inequality (3): If $a$ and $b$ are conjugate with respect to $K\triangleleft G$, and $b$ and $c$ are conjugate with respect to $L\triangleleft G$, then $a$ and $c$ are conjugate with respect to $KL\triangleleft G$. Since $|KL|$ divides $|K||L|$, we have $\nu_p(|KL|)\leq \nu_p(|K|)+\nu_p(|L|)$.
Define: $$d_t(a,b)=\min\{\nu_p(|H|)\mid H\triangleleft G, ab^{-1}\in H\}$$ Then again, $d_t$ has properties (1)-(4). Evidently, $d_t(a,b)\leq d_c(a,b)$ for every pair $a,b\in G$.
$d_c$ is not interesting in abelian groups where distinct elements are non-conjugate. Both $d_c$ and $d_t$ are not interesting in simple groups.
I came up with these metrics while reading about Alperin's fusion theorem, but I'm not aware of any link between that theorem and these metrics.
Some open-ended questions:
- Have these metrics, or something similar, been studied before?
- What interesting theorems can we prove about, or using, these metrics?
