Revision history [back]

There isn't a general construction of Frobenius groups in Sage currently, although there are many specific ones that can be constructed as members of other collections of groups, e.g. the symmetric group on 3 letters is a Frobenius group:

sage: G = SymmetricGroup(3)

Is it a Frobenius group? Let's check:

for g in G:
    if not g.is_one():
        fixed = [ x for x in [1..3] if g(x) == x ]  
        print "elt: ", g, "fixed set: ", fixed

returns:

elt:  (2,3) fixed set:  [1]
elt:  (1,2) fixed set:  [3]
elt:  (1,2,3) fixed set:  []
elt:  (1,3,2) fixed set:  []
elt:  (1,3) fixed set:  [2]

Yep.

I don't quite understand your second question. Any finite group can be embedded in a permutation group by its left or right action on itself. This is a basic theorem.