Frobenius group

A Frobenius group is a group G with a proper nontrivial subgroup H (the Frobenius complement) such that

H ∩ gHg−1 = {e} for every g ∉ H.

Equivalently G = K ⋊ H (a semidirect product) where H acts fixed-point-freely on K: only the identity of K is fixed by any nonidentity element of H. Then K is the Frobenius kernel.

Structure

• K ◁ G is normal; K = {e} together with all elements of G lying in no conjugate of H.
• K is nilpotent (Thompson's theorem) and equals the Fitting subgroup Fit(G).
• All complements H are conjugate.
• |H| divides |K| − 1, and |G| = |K|·|H|.

Examples

• The dihedral group D_n of order 2n for odd n is Frobenius, with cyclic kernel Cₙ and complement C₂.

See also

• semidirect product
• fitting — the kernel equals Fit(G)
• nilpotent — the kernel always is
• Frobenius group (Wikipedia)