25.73 Exponents

Exponents( U, u )
Exponents( U, u, F )

Exponents returns the exponent vector of an ag word u with respect to an induced generating system of U as list of integers if no field F is given. Otherwise the product of the exponent vector and F.one is returned. Note that u must be an element of U.

Let (u_1, ..., u_r) be an induced generating system of U. Then u can be uniquely written as u_1^{nu_1}* ...* u_r^{nu_r} for integer nu_i. The exponent vector of u is [nu_1, ..., nu_r].

Factor Groups of Ag Groups for details.

Note that Exponents adds a record component U.shiftInfo. This entry is used by subsequent calls with the same ag group in order to speed up computation. If you ever change the component U.igs by hand, not using Normalize, you must unbind the component U.shiftInfo, otherwise all following results of Exponents will be corrupted. In case U is a parent group you can use ExponentsAgWord (see ExponentsAgWord), which is slightly faster but requires a parent group U.

Note that you you may get a weird error message if u is no element of U. So it is strictly required that u is an element of U.

Note that Exponents uses ExponentsAgWord but not ExponentAgWord, so for records that mimic agwords Exponents may be used in ExponentAgWord.

    gap> v4 := AgSubgroup( s4, [ c, d ], true );
    Subgroup( s4, [ c, d ] )
    gap> Exponents( v4, c * d );
    [ 1, 1 ]
    gap> Exponents( s4 mod v4, a * b^2 * c * d );
    [ 1, 2 ] 

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