I am using the FreeGroup function in GAP
gap> f:=FreeGroup(2);; gap> Order(f); infinity gap> Order(f.1); # GAP Hangs here with not output # with Ctrl-c ^CError, user interrupt in MakeImmutable( a ); called from pow * obj called from <function "unknown">( <arguments> ) called from read-eval loop at line 18 of *stdin* you can 'return;' brk>
I quote from Wikipedia here
On the other hand, any nontrivial finite group cannot be free, since the elements of a free generating set of a free group have infinite order.
Order(f.1) should return
Infinity as well (for free group not for any general group)? I know that mathematically it is possible that one of them has a finite order and other has infinite order (for general group). Shouldn't my statement be true for "Free Group" specifically?