Question: how to implement a free group with symmetric properties

how to implement a free group with symmetric properties?

i find that equivalence relations = relexive, symmetric and transitive , is only these three ?

if implement symmetric properties, does it mean that adding equivalence relations to group?

how to implement free group with this?

 

i guess something like

f := FreeGroup( "a", "b" );
g := f / [equivalence relations....];

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