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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

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  • $\begingroup$ For GAP, you may see the GAP Software Carpentry lesson, in particular this episode: alex-konovalov.github.io/gap-lesson/05-small-groups $\endgroup$ – Alexander Konovalov Apr 27 at 11:53
  • $\begingroup$ thank you aleksander. that seems interesting. i will have a look. does it also describe how to compute the word length given a generating set? $\endgroup$ – orgesleka Apr 27 at 11:56
  • $\begingroup$ In GAP, Factorization provides a minimum length word expression, see math.stackexchange.com/questions/1962353/… $\endgroup$ – ahulpke Apr 27 at 14:50
  • $\begingroup$ @ahulpke: Thanks for your comment. This is very useful! I will have a look at the function and the related question. $\endgroup$ – orgesleka Apr 27 at 16:20

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