50.26 Permutation Character Candidates

For groups H, G with Hleq G, the induced character (1_G)^H is called the permutation character of the operation of G on the right cosets of H. If only the character table of G is known, one can try to get informations about possible subgroups of G by inspection of those characters pi which might be permutation characters, using that such a character must have at least the following properties:

:

pi(1) divides |G|,

:

[pi,psi]leqpsi(1) for each character psi of G,

:

[pi,1_G]=1,

:

pi(g) is a nonnegative integer for g in G,

:

pi(g) is smaller than the centralizer order of g for 1not= g in G,

:

pi(g)leqpi(g^m) for g in G and any integer m,

:

pi(g)=0 for every |g| not diving frac{|G|}{pi(1)},

:

pi(1) |N_G(g)| divides |G| pi(g), where |N_G(g)| denotes the normalizer order of < g > .

Any character with these properties will be called a permutation character candidate from now on.

GAP provides some algorithms to compute permutation character candidates, see PermChars. Some information about the subgroup can computed from a permutation character using PermCharInfo (see PermCharInfo).

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