By Leonard E Dickson

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**Extra resources for Algebraic invariants**

**Example text**

Then H is cyclic by induction, and so G has a cyclic subgroup of index p. 1, G is either cyclic, isomorphic with Modn (p), or is isomorphic with Cpn−1 × Cp . The last two are clearly not of p-rank 1, and so G must be cyclic, as required. For the case where p = 2, we start off with Philip Hall’s original theorem, about which we had a generalization of a special case earlier. The proof will be omitted. 5 (P. Hall) Let G be a p-group in which every characteristic abelian subgroup is cyclic. Then G is the central product of an extra-special group E and a p-group R, where R is either cyclic, dihedral, semidihedral, or quaternion.

Since [H, K] = 1, we see that L E, and similarly L G, proving that E = 1, and G is cyclic. Hence we reduce to the case where p = 2. 13, we see that E = Q8 . Thus G can only be the central product of Q8 and a cyclic, dihedral, semidihedral or quaternion subgroup. Let H be a normal subgroup of E of order 4, and let K be a normal subgroup of R of order 4, both containing the central element z = 1. Then HK has order 8 and exponent 4, since [H, K] = 1, and so HK is a non-cyclic, abelian normal subgroup, contrary to hypothesis.

Proof: Let x and y be elements of P , and suppose that there is g ∈ G such that xg = y. Thus Pg CG (x)g = CG (xg ) = CG (y). Thus both P and P g are Sylow p-subgroups of CG (y), whence they are conjugate by some element h ∈ CG (y). Thus P gh = P , and so gh ∈ NG (P ), and xgh = y, as required. As a corollary of Frobenius’ normal p-complement theorem, we get Burnside’s normal p-complement theorem. 7 (Burnside’s Normal p-complement Theorem) Let G be a finite group, and let P be a Sylow p-subgroup of G.