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Subject: Cube terms
From: mark.longridge@canrem.com (Mark Longridge)
Message-Id: <60.938.5834.0C1C7919@canrem.com>
Date: Fri, 6 Jan 1995 23:50:00 -0500
Organization: CRS Online  (Toronto, Ontario)

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 <-
Email: mark.longridge@canrem.com