Re: Product of orders of generators of a finite group

..> I have a question about finite groups:
..> is it true or false that, for each finite group G,
..> there is some set {g_1,...,g_n} of generators of G
..> such that the order of G is the product of the
..> orders of the g_i's?

I'm pretty sure the statement does hold for regular
p-groups; a group of order p^a with a<=p will
necessarily be regular.
And from regular p-groups you can go to solvable groups
via Hall's Theorems ...

Perhaps even just take generating sets of one Sylow per p.
The subgroup generated by those will have the same order
as the whole group. This may allow one to handle
insoluble groups, though not very many (I believe a 2-group
is regular if and only if it is abelian, and only a few
composition factors have abelian sylow 2-subgroups).

Is it obvious how to get a suitable generating set for a
regular p-group?

It is strange, but I think one cannot always choose the
generating set to have minimal cardinality. The extra-
special group of order 27 and exponent 3 can be generated
by three elements of order 3, but it can also be generated
by two elements of order 3, and has no element of order 9.

It might be that groups without such generating sets are
a little rare. Q8 is the only one of order 8, and Q8 x C2
is the only one of order 16. However, I think 7 of order
32 have no such generating set.

This gives a little flexibility for insoluble groups,
since a few more composition factors are allowed.
.