58.3 Operations for Automorphism Group Elements

a = b

For aut group elements a and b, the operator = evaluates to true if the aut records correspond to the same aut, and false otherwise. Note that this may return true even when the two records themselves are different (one of them may have more information stored in it).

a * b

For aut group elements a and b, the operator * evaluates to the product a b of the auts.

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a / b

For aut group elements a and b, the operator / evaluates to the quotient a b^{-1} of the auts.

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a ^ i

For an aut group element a and an integer i, the operator ^ evaluates to the i-th power a^i of a.

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a ^ b

For aut group elements a and b, the operator ^ evaluates to the conjugate b^{-1} a b of a by b.

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Comm(a, b)

The function Comm returns the commutator a^{-1} b^{-1} a b of the two aut group elements a and b.

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g ^ a

For a sag group element g and an aut group element a, the operator ^ evaluates to the image g^a of the ag word g under the aut a. The sag group element g must be an element of a.group.

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S ^ a

For a subgroup S of a sag group and an aut group element a, the operator ^ evaluates to the image S^a of the subgroup S under the aut a. The subgroup S must be a subgroup of a.group.

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list * a
a * list

For a list list and an aut group element a, the operator * evaluates to the list whose i-th entry is list[i] * a or a * list[i] respectively.

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list ^ a

For a list list and an aut group element a, the operator ^ evaluates to the list whose i-th entry is list[i] ^ a.

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Note that the action of aut group elements on the elements of the sag group via the operator ^ corresponds to the default action OnPoints (see Other Operations) so that the functions Orbit and Stabilizer can be used in the natural way. For example:

inputOut.Orbit

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