Martin Schoenert states:
Only one out of 332640 elements of GE (and of G) centralizes P.
That is to say that the index of the centralizer of P in GE has index
332640 in GE. Since all elements of GC commute with all elements of
GE, the index of the centralizer of P in G also has index 332640 in G.
Z is indeed the center of GE', GE, G, G', and GCE.
I get the fact that only the super-flip (or 12-flip) is the centre of
G and the centre of GE. Another way to look at it would be the centre
of the cube group must effect all the corners & edges in the same
way, and only the super-flip fits these conditions when we allow
all 6 generators < U, D, F, B, L, R > to be used.
In the case of the smaller group < U, R > we can get 6 corners twisted
either clockwise or counter-clockwise, thus effecting all the corners
and edges the same, due to the fact we can have 6 twists the same and
< U, R > only contains 6 corners, and so this is the centre of
< U, R >.
But I don't understand how only one out of 332,640 elements of GE
and G centralizes P. I thought that GE had:
(12 ^ 2 / 2 ) * 12! = 980,995,276,800 elements
That is to say that the group on the cube of edges only has
980,995,276,800 elements. To be honest I'm not sure what P
represents! Jerry refers to P as the Pons Asinorum, but I think
the term may have two meanings in the two messages.
Z is the centre of G right? I need an ANSI standard math dictionary,
but I doubt such a book exists.
I'm going to tackle some more cube terminology in my next message.
-> Mark <-