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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:

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Dear David

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.

Best wishes

Marston Conder

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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

cases.

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Finally as to the technical problem that caused Marston's letter to be

withheld;

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

Mike

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