# 7.3 Operations for Group Elements

`g * h` `g / h`

The operators `*` and `/` evaluate to the product and quotient of the two group elements g and h. The operands must of course lie in a common parent group, otherwise an error is signaled.

`g ^ h`

The operator `^` evaluates to the conjugate <h>^{-1}* <g>* <h> of g under h for two group elements elements g and h. The operands must of course lie in a common parent group, otherwise an error is signaled.

`g ^ i`

The powering operator `^` returns the i-th power of a group element g and an integer i. If i is zero the identity of a parent group of g is returned.

`list * g` `g * list`

In this form the operator `*` returns a new list where each entry is the product of g and the corresponding entry of list. Of course multiplication must be defined between g and each entry of list.

`list / g`

In this form the operator `/` returns a new list where each entry is the quotient of g and the corresponding entry of list. Of course division must be defined between g and each entry of list.

`Comm( g, h )`

`Comm` returns the commutator <g>^{-1}* <h>^{-1}* <g>* <h> of two group elements g and h. The operands must of course lie in a common parent group, otherwise an error is signaled.

`LeftNormedComm( g1, ..., gn )`

`LeftNormedComm` returns the left normed commutator `Comm( LeftNormedComm( g1, ..., gn-1 ), gn )` of group elements g1, ..., gn. The operands must of course lie in a common parent group, otherwise an error is signaled.

`RightNormedComm( g1, g2, ..., gn )`

`RightNormedComm` returns the right normed commutator `Comm( g1, RightNormedComm( g2, ..., gn ) )` of group elements g1, ..., gn. The operands must of course lie in a common parent group, otherwise an error is signaled.

`LeftQuotient( g, h )`

`LeftQuotient` returns the left quotient <g>^{-1}* <h> of two group elements g and h. The operands must of course lie in a common parent group, otherwise an error is signaled.

GAP 3.4.4
April 1997