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Let $G$ be a finite group. We say that the lattice of subgroups of $G$ is graded if it is possible to assign a non-negative integer rank $r(H)$ to each subgroup $H$ in such a way that the following two properties hold.

  • If $H \leq K$, then $r(H) \leq r(K)$.
  • If $H \leq K$ and no group is strictly between $H$ and $K$, then $r(K)=r(H)+1$.

Examples of groups with graded lattices include

  • Abelian groups ($r(H)$ is the number of prime factors in $|H|$, counted with multiplicity).
  • The symmetric group $S_3$ (the identity has rank $0$, the whole group has rank $2$, all other subgroups have rank $1$)
  • The quaternion group (identity has rank $0$, $\{1, -1\}$ has rank $1$, the subgroups generated by $i$, $j$, and $k$ have rank $2$, and the whole group has rank $3$).

An example of a group without a graded lattice is $A_4$, the alternating group on $4$ elements. One way to see this is to note that you have the two chains $$\{e\} \leq \{e, (12)(34)\} \leq \{e, (12)(34), (13)(24), (14)(23)\} \leq A_4$$ and $$\{e\} \leq \{e, (123), (132)\} \leq A_4$$ The two chains are both maximal ($A_4$ has no subgroups of order $6$), but looking at the two chains give conflicting values for the rank of the whole group vs. that of the identity.

Is there a general description of which groups do or do not have graded subgroup lattices?

I feel like this should have been studied somewhere before, but can't find anything on it.

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    $\begingroup$ I think that these groups (i.e. with graded subgroup lattice) are precisely the supersoluble groups and that the result is due to Iwasawa. $\endgroup$ – the_fox Apr 4 '19 at 22:08
  • $\begingroup$ I guess "catenary" would be a better terminology (it's used in ring theory for basically the same meaning, while "graded" is widely used for unrelated concepts). $\endgroup$ – YCor Apr 5 '19 at 23:00
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Thank you the_fox for bringing Iwasawa's result to my attention. This answer is expanding on their comment.

Call a group $G$ Supersolvable (or Supersoluble) if there is a chain of subgroups

$$e=H_0 \vartriangleleft H_1 \dots \vartriangleleft H_m = G$$

Such that each $H_i$ is normal in $H_{i+1}$ and each quotient $H_{i+1}/H_i$ is cyclic (as opposed just being Abelian as required by regular solvability). There are a few characterizations of these groups, but one (due to Iwasawa in 1941's "Über die endlichen Gruppen und die Verbände ihrer Untergruppen" (MathReview, paper doesn't seem to be available online) is that these are the groups whose maximal chains of subgroups all have the same length, which is equivalent to having a well-defined rank function.

Hall's book The Theory of Groups devotes section 19.3 and portions of section 10.5 to this theorem and its proof. 10.5 also mentions another interesting characterization of these groups due to Huppert: A group is supersolvable if and only if all maximal proper subgroups have prime index.

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