Dear Fatemeh:
Lemma 1 of Bertram's 1991 paper in the Israel Journal of Math, vol 75
(1991) pp. 243-245 offers a characterisation of Frobenius groups which
is easy to implement and computationally fast. In summary it states:
G is a group with a proper normal subgroup N satisfying k(G) = k(G/N) +
(k(N) - 1)/[G:N], if and only if G is a Frobenius group with kernel N
(in which case N is the Fitting subgroup of G).
Here k(H) denotes the number of conjugacy classes of a group H.
Best wishes.
Eamonn
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