Let $G$ be a finite group, $S \subset G$ a generating set. Set $\sigma(G):=\sum_{U \subset G} |U| $, where the sum runs over all subgroups $U$ of $G$. Set $H_G := \sum_{g \in G} \frac{1}{|g|+1}$, where $|g|:= $ word length (with respect to $S$). For $G:=\mathbb{Z}/(n)$ we get $\sigma(G) = \sigma(n)=$ sum of divisors of $n$. and $H_{\mathbb{Z}/(n)} = H_n=n$-th harmonic number, where $S=\{+1\}$. My naive conjecture inspired by Lagarias inequality is $$ \sigma(G) \le H_G + \exp(H_G) \log(H_G)$$
For $G:=\mathbb{Z}/(n)$ and $S:=\{+1\}$ this is the Lagarias inequality. I have checked in Sagemath for the symmetric group up to $n=6$:
def sigmaGr(G):
return sum([len(U.list()) for U in (G.subgroups())])
def wordLen(g):
return g.length()
def HG(G):
return sum([1/(wordLen(g)+1) for g in G.list()])
def LG(G):
H = HG(G)
return (H+exp(H)*log(H)).N()
for n in range(1,6):
G = SymmetricGroup(n)
print sigmaGr(G),LG(G)
My question is, how to compute if the inequality is true for some small groups in SAGEMATH or GAP GROUP THEORY?
Thanks for your help!
Related: https://mathoverflow.net/questions/330077/a-group-theoretic-interpretation-of-lagarias-inequality
Factorization
provides a minimum length word expression, see math.stackexchange.com/questions/1962353/… $\endgroup$