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permutation representation

asked 2013-01-27 07:29:16 +0100

Babgen gravatar image

updated 2013-01-27 07:30:05 +0100

Dear all

Does sage support groups such as Frobenius groups or general affine group. How can I find the permutation representation of an arbitrary group.

Best regrad

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answered 2013-01-28 01:29:47 +0100

benjaminfjones gravatar image

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


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]


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.

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Asked: 2013-01-27 07:29:16 +0100

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Last updated: Jan 28 '13