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?
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asked Dec 21, 2015 at 13:58
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$\begingroup$ math.stackexchange.com/questions/1566605 is relevant - or math.stackexchange.com/questions/1004210 $\endgroup$– Derek HoltCommented Dec 21, 2015 at 16:12
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$\begingroup$ 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). $\endgroup$– Yiftach BarneaCommented Dec 21, 2015 at 16:18
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$\begingroup$ 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 $4$. $\endgroup$– Derek HoltCommented Dec 21, 2015 at 16:27
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$\begingroup$ 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. $\endgroup$– Yiftach BarneaCommented Dec 21, 2015 at 17:01
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1$\begingroup$ See the paper by Lucchini The largest size of a minimal generating set of a finite group $\endgroup$– M. Farrokhi D. G.Commented Dec 22, 2015 at 1:00
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2 Answers
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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.
answered Dec 24, 2015 at 0:56
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$\begingroup$ 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? $\endgroup$ Commented Dec 28, 2015 at 8:39
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1$\begingroup$ 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 $\mu(L):=m(L)-m(L/N)$. $\endgroup$ Commented Dec 29, 2015 at 1:59
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1$\begingroup$ In the second paper, lower and upper bounds are given for the quantity $\mu(L)$ when $L$ is a monolithic group. $\endgroup$ Commented Dec 29, 2015 at 2:00
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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?
answered Dec 21, 2015 at 15:26
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$\begingroup$ 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 $G$. $\endgroup$– SevaCommented Dec 21, 2015 at 15:47
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$\begingroup$ 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. $\endgroup$ Commented Dec 21, 2015 at 15:53
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1$\begingroup$ 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 $D(G)=D(G')=k$. $\endgroup$– YCorCommented Dec 22, 2015 at 10:06