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Chris Wensley writes in his e-mail message of 1994/04/13

I have a query about how GroupHomomorphismByImages works.

In the listing below Q,P are copies of D4 with Q an fp-group

and P a permutation group. When mapping Q to P there is only

one way to express the images of the generators. In the reverse

direction the image of generator (1,2,3,4) is listed

(surprisingly to me) as f.2^-1*f.1^-2*f.2^-1 rather than f.1

(although, of course, these are equal).

Here is what happens if you have a 'GroupHomomorphismByImages' from a

permutation group $P$. First GAP computes the size of $P$. Then it

constructs a (second) stabilizer chain for $P$, which is stored in the

record that represents the homomorphism (see "Stabilizer Chains" in the

manual for a explanation of what a stabilizer chain is).

Each computation is performed in parallel also with the images of the

generators. That means that the stabilizer chain contains for each

strong generator $s_i$ also its image under the homomorphism $t_i$.

To compute the image of an arbitrary element $p$ in the permutation

group, this element is divided through the stabilizer chain, i.e.,

'Image' computes a word $w$ in the strong generators $s_i$, such that

$p w(s_i) = ()$. In parallel we perform the same computation with the

images of the strong generators, i.e., we compute $w(t_i)$, which is

the inverse of the image.

Now when 'GroupHomomorphismByImages' constructs the stabilizer chain

it works essentially random. That is, it constructs random elements

and builds the stabilizer chain from those. It stops when the chain

is complete, i.e., when the product of the indices in the stabilizer

chain is equal to the size of the group $P$.

This helps to keep down the length of the words somewhat (but not enough

in most cases). On the other hand it makes it impossible to predict what

word you get when you map from $P$ into the free group $F$ (or a factor

thereof, but the words are actually always in $F$).

He continues

(Sorry if this is a trivial query, but I am a new user.)

There would be no need to apologize even if the question was trivial,

which it is not.

Hope this helps (if not, don't hesitate to ask again).

Martin.

-- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .- Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551 Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany

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