# Maximal size of minimal generating set

Let $G$ be a finite group. Denote by $D(G)$ the maximal size of a minimal generating set, (minimal in the sense of inclusion). I vaguely remember seeing recently something on $D(G)$. Can anyone refer me to anything new or old?

• – Derek Holt Dec 21 '15 at 16:12
• Since people asked below let me say: I am interested in an absolute upper bound for $D(G)$ for particular sequences of groups. So I do not care about the exact value of $D(G)$. Also I understand the situation of nilpotent groups. Also notice that $D(G \times H) \geq D(G)+D(H)$ (possibly it is equal). – Yiftach Barnea Dec 21 '15 at 16:18
• But then you should ask about the specific sequences of groups that you are interested in. As it stands, the question is too broad to allow a useful answer. As you can see from the second link above, this number has been determined for ${\rm PSL}(2,p)$ - it is $3$ or $4$. – Derek Holt Dec 21 '15 at 16:27
• Derek, I don't know exactly what I am looking for that is why I have asked for reference. But your answer might be of interest. – Yiftach Barnea Dec 21 '15 at 17:01
• See the paper by Lucchini The largest size of a minimal generating set of a finite group – M. Farrokhi D. G. Dec 22 '15 at 1:00

The maximum size of minimal generating sets of a finite group is studied recently by Lucchini in the following two papers:

Andrea Lucchini, The largest size of a minimal generating set of a finite group, Arch. Math. (Basel) 101(1) (2013), 1–8.

Andrea Lucchini, Minimal generating sets of maximal size in finite monolithic groups, Arch. Math. (Basel) 101(5) (2013), 401–410.

• These don't seem to be available of Open Access, and the abstracts don't seem to describe the results in detail. Is the outcome an algorithm for determining $D(G)$, or is there an explicit general formula? – Geoff Robinson Dec 28 '15 at 8:39
• For solvable groups $m(G)$ (the notation used in these and other papers for $D(G)$) is simply the number of non-Frattini factors in a chief series of $G$. For an arbitrary group $m(G)=\sum_A\mu(L_G(A))$, where $A$ ranges over non-Frattini factors in a chief series of $G$. Here $L_G(A)$ is the monolithic primitive group associated with $A$ defined as $L_G(A)=A\rtimes G/C_G(A)$ if $A$ is abelian and $L_G(A)=G/C_G(A)$ if $A$ is non-abelian. Also, for a monolithic group $L$ with the socle $N$, $\mu(L)$ is defined as $\mu(L):=m(L)-m(L/N)$. – M. Farrokhi D. G. Dec 29 '15 at 1:59
• In the second paper, lower and upper bounds are given for the quantity $\mu(L)$ when $L$ is a monolithic group. – M. Farrokhi D. G. Dec 29 '15 at 2:00

If $|G|= \prod_{i=1}^{r} p_{i}^{n_{i}}$ where the $p_{i}$ are distinct primes and each $n_{i}$ is a positive integer, then any minimal generating set of $G$ has at most $\sum_{i=1}^{r}n_{i}$ elements, and this bound can be attained whenever $G$ is Abelian of squarefree exponent. This is rather elementary, but what sort of improvement were you hoping for?

• What if $G$ is not abelian of square-free exponent? As I understand if, the PO wants to know $D(G)$ precisely for all finite groups $G$. – Seva Dec 21 '15 at 15:47
• I read the question differently, but the PO may clarify. I think it is a rather tall order to expect to know $D(G)$ precisely for all finite groups, but I am prepared to be proved wrong. – Geoff Robinson Dec 21 '15 at 15:53
• An improvement is to precisely give $D(G)$ for any abelian (or nilpotent) $G$. Namely, let $G$ be nilpotent and let $G'$ be its quotient by its Frattini subgroup, so $G'$ is a product of $k$ cyclic groups of prime order. Then $D(G)=D(G')=k$. – YCor Dec 22 '15 at 10:06