> < ^ From:

< ^ Subject:

Dear GAP-Forum,

George McNinch wrote:

Except that GAP does seem to be able to decide invertibility:

gap> R := Integers mod 9; (Integers mod 9) gap> S := FullMatrixAlgebra(R,2); ( (Integers mod 9)^[ 2, 2 ] ) gap> one := Identity(R); zero := Zero(R); ZmodnZObj( 1, 9 ) ZmodnZObj( 0, 9 ) gap> x := [[one,one],[zero,one]]; [ [ ZmodnZObj( 1, 9 ), ZmodnZObj( 1, 9 ) ], [ ZmodnZObj( 0, 9 ), ZmodnZObj( 1, 9 ) ] ] gap> IsUnit(S,x); true

For this particular element x, but not for all (in particular if matrix

entries are zero divisors): With

gap> l:=AsList(S);;

l[28] cannot be inverted in GAP 4.2. (The matrix inversion routine assumes

all nonzero entries will be invertible.)

> And one can even generate SL(2,Z/9Z):

>

> gap> y := [[one,zero],[one,one]];

> [ [ ZmodnZObj( 1, 9 ), ZmodnZObj( 0, 9 ) ],

> [ ZmodnZObj( 1, 9 ), ZmodnZObj( 1, 9 ) ] ]

> gap> Order(Group(x,y));

> 648

Since the generators are invertible GAP forms a group. However there still

are group elements which (in GAP 4.2) for the same reasons cannot be

inverted.

Without doubt these will cause problems for further calculations.

(As mentioned before, it will work in the next release.)

Best wishes,

Alexander Hulpke

-- Colorado State University, Department of Mathematics,

Weber Building, Fort Collins, CO 80523, USA

email: hulpke@math.colostate.edu, Phone: ++1-970-4914288

http://www.math.colostate.edu/~hulpke

> < [top]