--- In [email protected], "d_j_salvia" <[EMAIL PROTECTED]>
wrote:
>
> Hi Everyone,
>
> What is the definition of a "generator"?
>
> Thank you,
>
> David J
>
In a group, G, a set of generators is any set whose closure under the group
operations
(multiplication and inverse) is the entire group. For a finite group this is
equivalent to the
closure under multiplication being the whole group (as the inverse of an
element is some
power of that element). By convention the trivial group is generated by the
empty set.
(Because the identity element is the empty product.)
If A is a set of generators then so is any superset of A (in G).
A generator is an element of a set of generators. It only really makes sense to
talk about a
generator within the context of a set of generators. Typical sets of generators
include
{U,D,F,B,R,L} and {U,U2,D,D2,F,F2,B,B2,R,R2,L,L2} although neither of these is
a minimal
set of generators.
For instance U is a generator if the set of generators is {U,D,F,B,R,L} but not
in {D,F,B,R,L}
(which also generates the usual cube group, though not of course the supercube
group).
U2 is not a generator in the QTM generators but it is in the HTM generators.
A set of generators is minimal if it contains no proper subset which is a set
of generators.
This is minimal in the lattice of subsets of G - but it doesn't mean that such
a set has
minimal size amongst sets of generators (though a set of generators of minimal
size is
obviously a minimal set of generators).
Any group has a set of generators - you can take the set of all the elements of
the group
as a trivial example (and you can also exclude the identity so an group has at
least 2 sets
of generators). At least assuming the axiom of choice every group has a minimal
set of
generators too. (I'm not sure if that can be proven without AC - it's been a
long time...)
Any generating set for the Rubik cube group has at least 2 elements (because a
group with
a generating set of fewer than 2 elements is Abelian, that is to say it
commutes) and
indeed the Rubik cube group is a 2-generator group, which is to say that there
is a
generating set with 2 elements.
The supercube group is a 6-generator group. There's an obvious set of
generators of size
6, but by considering the action on the centres of any generating set we see
that the
generating set must have size at least 6. (Basically, the action on the centres
must be
commutative with each generator having order dividing 4 (wrt it s action on the
centres)
which means that a set of 5 elements can generate at most 4^5=1024 possible
centre
positions, whereas tere are 2048).
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