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GAP-Forum,

This is in response to:

From: Phil Meier <phil_meier99@hotmail.com>

Dear Forum,

I have explicitly given a finitely generated

(non-abelian) free group G,

a finite index subgroup H,

and a finite index normal subgroup of H, called N.

(N is is in general not normal in G.)

(H and N are given as finitely generated subgroups of G).Now my question:

Can GAP check if the intersection of H with

the normal closure of N in G is equal to N?Thanks

Phil

Here is one way to do this, assuming you start out with lists of

generators for H and N. I forget the exact GAP commands, but they

shouldn't be too hard to figure out.

1. Do a coset enumeration of the finitely presented group < G | N >,

i.e., the (finite) group with generators the generators of G, and

relators the generators of N. This will give you the action of G

on the cosets of N^G (the normal closure of N in G). Call this

domain D.

2. Using the generator list for H, construct the corresponding image Q

of H in Sym(D), i.e., the image Q of H arising from the induced

action of H on the cosets of N^G. The order of Q will be the index

of (the intersection of H and N^G) in H, which you can then compare

with the index of N in H to see if they are equal.

Tim Hsu

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