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The Base Group of a Finite Wreath Product PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000{000 S 0002-9939(XX)0000-0 THE BASE GROUP OF A FINITE WREATH PRODUCT BEN BREWSTER, D. S. PASSMAN, AND ELIZABETH WILCOX Abstract. Let W = G o H denote the permutational wreath product of the finite group G by H ⊆ Symn, and suppose that H is transitive on f1; 2; : : : ; ng or at least that it acts faithfully on all its orbits. Our main result determines all those wreath products W which have a non-characteristic base subgroup B = Q G. In addition, we briefly discuss those automorphisms of W that move this base. 1. Introduction In the following, let G and H be finite groups with H ⊆ Symn, and we let W = G o H denote the permutational wreath product of G by H. Then W has a normal subgroup B = Q G, its base group, which is a direct product of n copies of G, and W = B o H where H acts on B by naturally permuting its n factors. This wreath product construction provides a rich source of examples of finite groups and intuition regarding general group-theoretic structures. For example, [Hu, Satz I.15.9] shows that any extension group can be embedded in a suitable wreath product. Since these groups occur so frequently, it is certainly of interest to determine their automorphism groups. Indeed, there is a rich literature on this subject when the base subgroup B is characteristic in W . See for example [B], [Ha], and [M]. The question of when the base group is characteristic has also been considered. For simplicity, we state the following two results in the context of finite groups. To start with, we say that a group is odd-dihedral if it is the split extension A o hxi, where A is an abelian group of odd order and where x has order 2 and acts on A by inverting all its elements. Furthermore, the wreath product G o H is said to be standard if H is a regular subgroup of Symn. Theorem 1.1. [N, Theorem 9.1]. Let W = G o H be a standard wreath product with G and H finite. Then the base B = Q G is characteristic in W unless G is ∼ an odd-dihedral group, n = 2, and H = Sym2 = C2. Under less stringent hypotheses, there is the somewhat weaker Theorem 1.2. [B, Theorem 1]. If the base group of the wreath product W = G o H is not characteristic in W , then G is either an elementary abelian 2-group or a dihedral group, and H has a non-trivial normal elementary abelian 2-subgroup. Actually, the proof of Theorem 1.2 seems to be incomplete, and indeed there is some confusion in the mathematical community over this issue. While the result is stated again in [KBF] without proof, [M, page 101] asserts that no such general 2000 Mathematics Subject Classification. 20E22, 20E36. c XXXX American Mathematical Society 1 2 BEN BREWSTER, D. S. PASSMAN, AND ELIZABETH WILCOX result is known. In this paper, we settle the question definitively in the case of finite groups. Specifically, we prove Theorem 1.3. Let W = G o H be the wreath product of the finite group G by H ⊆ Symn and assume that H is transitive or at least that it acts faithfully on each orbit. Then the base group B = Q G fails to be characteristic in W if and only if G is an odd-dihedral group and H = C2 o Y , where C2 = Sym2, n is even, and Y ⊆ Symn=2. In this case, the base group of W = G o H = G o (C2 o Y ) = (G o C2) o Y Q given by N = (G o C2) is characteristic in W . This will be proved in Section 3. Note that the latter fact about N being char- acteristic allows one to compute Aut(W ) by switching from the non-characteristic base B of W = G o H to the characteristic base N of W = (G o C2) o Y . Much of this research was begun during the second author's recent visit to the Mathematics Department of SUNY-Binghamton. The results of this paper will constitute part of the third author's forthcoming dissertation. 2. Arbitrary Actions Again, G and H are finite groups with H ⊆ Symn, and W = G o H denotes the permutational wreath product of G by H. In particular, W is the split extension Q W = B o H where the base subgroup B = G is a direct product of n copies of G and where H acts on B by naturally permuting its n factors. For convenience, we write g = jGj and we say that x is a coordinate element of B if it is contained in precisely one of the G direct factors. Obviously, B is generated by its coordinate elements. Of course, if either G or H is the identity group, then B = h1i or W is clearly characteristic in W . Thus it suffices to assume throughout that G and H are both nonidentity groups. As a consequence, H acts faithfully on B, and indeed H =∼ W=B acts faithfully as a permutation group on the G factors of B. Our first goal is to show that O2(B), the smallest normal subgroup of B with factor group a 2-group, is characteristic in W . To this end, for any x 2 W , let κ(x) denote the number of W -conjugates of x that do not commute with x, and set K = fx 2 W j κ(x) < gg: Obviously, K is a characteristic subset of W . Lemma 2.1. With the above notation, we have i. If x is a coordinate element of B, then x 2 K. ii. If x 2 K, then x2 2 B. Proof. (i) If x is a coordinate element of B, then clearly all W -conjugates xw of x are also coordinate elements. If x and xw lie in different factors of B, then x and xw commute. Thus the conjugates counted in κ(x) are all contained in a single factor G containing x. Indeed, this subset is then contained in G n fxg and has size at most g − 1. (ii) Suppose x has a t-cycle, say (1 2 ··· t), in its permutation action on the factors of B, and let this correspond to the direct factor T of B given by T = G1 × G2 × · · · × Gt. If u 2 CT (x), then the G1-entry of u uniquely determines u. Indeed, if u = (u1; u2; : : : ; ut) 2 T and if x = (x1; x2; : : : ; xn)τ 2 BH, then THE BASE GROUP OF A FINITE WREATH PRODUCT 3 x xt x1 x2 xt−1 x1 u = (ut ; u1 ; u2 ; : : : ; ut−1 ), and thus u 2 CT (x) if and only if u2 = u1 ; u3 = x2 xt−1 xt u2 ; : : : ; ut = ut−1 and u1 = ut . Thus jCT (x)j ≤ g and hence x has jT : CT (x)j ≥ gt−1 conjugates under T . Now if v 2 T , then xv = x[x; v] and the commutator [x; v] is contained in T since T/ hT; xi. Thus the set of commutators [x; T ] has size ≥ gt−1. Furthermore, x commutes with xv if and only if it commutes with [x; v]. It follows that conjugating x by elements of T gives rise to at least [x; T ] n CT (x) elements counted in κ(x), and hence κ(x) ≥ gt−1 − g. Finally, if t ≥ 3, then gt−1 − g ≥ g2 − g ≥ g, since g ≥ 2. Thus κ(x) ≥ g, contradicting the fact that x 2 K. We conclude that in its permutation action on the factors of B, x can only be a product of 2-cycles. In particular, x2 acts trivially 2 and hence x 2 B. With this, we can prove Proposition 2.2. If B is the base group of the wreath product W = G o H, then 2 Q 2 O (B) = O (G) is characteristic in W . Proof. Let S = hx2 j x 2 Gi. Then S/G with G=S being an elementary abelian 2-group. Now define K = hx2 j x 2 Ki, so that K is a characteristic subgroup of W . By part (ii) of the previous lemma, we see that K ⊆ B, and by part (i), we have K ⊇ S ×S ×· · ·×S. Thus, Q S ⊆ K ⊆ B = Q G, and since G=S is a 2-group, 2 2 so is B=K. It follows that O (B) = O (K) is characteristic in W . We will sharpen this result below. To start with, recall that a group is odd- dihedral if it is of the form A o hxi, where A is an abelian group of odd order and where x has order 2 and acts on A by inverting all its elements. The two results stated below are standard group-theoretic facts. We include their elementary proofs for the convenience of the reader. Of course, part (i) is a special case of the existence of Frobenius kernels. Lemma 2.3. Let D be a finite group. i. If D contains a self-centralizing subgroup of order 2, then D is odd-dihedral. ii. If more than 3=4 of the elements of D have order ≤ 2, then D is an ele- mentary abelian 2-group. Proof. (i) Let x be an element of order 2 in D with hxi self-centralizing. If X denotes the conjugacy class of x, then jXj = d=2, where d = jDj.
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