# A group theoretic interpretation of Lagarias inequality

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?

• In GAP, Factorization provides a minimum length word expression, see math.stackexchange.com/questions/1962353/… – ahulpke Apr 27 at 14:50