Dear Forum members,
This is my first email to the Forum in the role of its overseer. I'd like
to begin it by thanking Joachim for looking after the Forum so well until
now (and many other GAP-related activities as well!). The remainder of
this letter also owes much to Joachim.
The last question of David Joyner received a very nice and complete
answer in the letter of Jim Howie. However even before Jim Howie's
letter, also Marston Conder had sent a letter to the Forum, that
unfortunately was not posted by 'miles' for a technical reason, of
which I want to remind you at the end of this note. However first I
want to post Marston Conder's reply mainly because it mentions an
alternative way of showing that David Joyner's presentation does not
define M_23 which just uses GAP without any of the harder theory that
Jim Howie applies and since moreover it can often help in cases where
a presentation is not suitable for the application of such theory for
special types of presentation.
Marston had written:
This suggests M23 might be Free(a,b)/[(a*b)^6,(a^5*y^7)^4].
It's not quite so easy as that!
I think you'll find Free(a,b)/[(a*b)^6,(a^5*b^7)^4] has quite a few subgroups of index 2, 3, 4, etc, so can't be M23. In fact the subgroup generated by u = a^2, v = b and w = a*b*a has index 2 and presentation < u,v | (u*v*w)^3 = (u^3*v^7*u^2*w^7)^2 = 1 >, with infinite abelianisation, and therefore your group is infinite.
I just append a simple GAP run that does what Marston suggests, namely
computes all subgroups of index 2 and 3 and abelianizes them:
gap> F := FreeGroup( "a", "b" );; gap> a := F.1;; b := F.2;; gap> G := F / [ (a*b)^6, (a^5*b^7)^4 ]; Group( a, b ) gap> lis := LowIndexSubgroupsFpGroup( G, TrivialSubgroup( G ), 3 ); [ Group( a, b ), Subgroup( Group( a, b ), [ a, b*a*b^-1, b^2 ] ), Subgroup( Group( a, b ), [ a, b^2*a^-1*b^-1, b*a^2*b^-1, b*a*b ] ), Subgroup( Group( a, b ), [ b, a^2, a*b*a^-1 ] ), Subgroup( Group( a, b ), [ b, a*b*a^-2, a^3, a^2*b*a^-1 ] ), Subgroup( Group( a, b ), [ b*a^-1, a^2, a*b ] ), Subgroup( Group( a, b ), [ b*a^-1, a*b*a^-2, a^3, a^2*b ] ) ] gap> List( lis, H -> Index( G, H ) ); [ 1, 2, 3, 2, 3, 2, 3 ] gap> List( lis, H -> AbelianInvariantsSubgroupFpGroup( G, H ) ); [ [ 2, 3, 8 ], [ 0, 3, 4 ], [ 2, 3, 3, 4, 8 ], [ 0, 3, 4 ], [ 2, 3, 3, 8, 8 ], [ 2, 2, 3, 4 ], [ 2, 4, 4, 8 ] ]
The command LowIndexSubgroupsFpGroup can be used quite often in such
Finally as to the technical problem that caused Marston's letter to be
Some time ago we encountered several times the problem that letters
that were meant as private answers to letters in the GAP Forum were
sent off using 'reply' and then ended up in the Forum for which they
were not meant. We therefore at that time taught 'miles' to 'look
critically' at letters which were sent to the Forum by use of 'reply'
before sending them off to all Forum members. Miles will only do this
if such letters start with a phrase that contains the word 'GAP' or
the word 'Forum'. So if you start your reply with 'Dear Forum members'
or the like your letter will be sent to all Forum members, but if it
starts, as was the case with Marston's 'Dear David', miles suspects
that this might be a private letter to David Joyner and will send it
back (in this case to Marston) asking him to put in some magic word as
explained above in case the letter is really meant for the full Forum
and to resend it. Marston has obviously not done this because in the
meantime Jim Howie's letter had fully answered the question. However
both because of the useful suggestion in Marston's letter and because
of the opportunity to remind you of the way GAP replies are handled,
I thought it worth repeating this information.
Hope all this is useful