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Covering Groups of Rank 1 of Elementary Abelian Groups

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Covering Groups of Rank 1 of Elementary Abelian Groups Covering Groups of Rank 1 of Elementary Abelian Groups R. Gow and R. Quinlan Abstract: Covering groups of elementary abelian groups of odd exponent p can be classified according to the rank of their p–th power homomorphisms, which may be regarded as linear transformations of Fp–vector spaces. This paper contains a description of the isomorphism types and the automorphism groups of those covering groups in which this rank is 1. Analogous considera- tions of elementary abelian 2–groups and their covering groups are included in the final section. Let p be an odd prime, let n be a positive integer and let Q denote the elementary abelian group of order pn. In this paper we associate to every covering group of Q its rank, which is an integer in the range 0, . , n. We begin by constructing a group H of order pn(n+1)/2 which has every covering group of Q as a central quotient. In Section 1 we describe the automorphism group of H and use it to derive a formula of P. Hall involving the orders of the automorphism groups of all the non–isomorphic covering groups of Q. In Section 2 we define the rank of a covering group and prove a modified version of Hall’s formula for covering groups of fixed rank. Sections 3 and 4 are devoted to a detailed investigation of the rank 1 case, using tools from linear algebra. In Section 5 we consider related questions about elementary abelian 2–groups and their covering groups, but here the information available is combinatorial rather than algebraic in nature. We begin with the basic definitions. For any group G and elements x, y of G, we let [x, y] denote the commutator x−1y−1xy. As usual we denote the commutator subgroup and centre of G by G0 and Z(G) respectively. Suppose for now that p is any prime and n is a positive integer and let Q be an elementary abelian p–group of order pn. A group Q˜ is said to be a covering group (or stem cover) of Q if Q˜ contains a subgroup Z with the following properties : • Z ⊆ Z(Q˜) ∩ Q˜0; • Z is an elementary abelian group of order pn(n−1)/2; • Q/Z˜ =∼ Q. 1 Any such covering group Q˜ is a central quotient of a larger finite p–group H whose structure allows us to investigate the properties of all the covering groups in a uniform way. The group H is defined as follows, see also [3]. Let F be a free group of rank n with free generators a1,..., an. Let R be the fully invariant subgroup of F generated by 2 xp , [x, y]p, [xp, y] and [[x, y], z] for all x, y and z in F , and let H = F/R. H is a finite p–group generated by the n elements hi = aiR for 1 ≤ i ≤ n. Let Φ(H) denote the Frattini subgroup of H. This is clearly a central subgroup of H. Higman shows in [3] that Φ(H) is an elementary abelian p–group of n(n+1)/2 p p order p generated by the n(n+1)/2 independent elements h1,..., hn and [hi, hj] for 1 ≤ i < j ≤ n. Let C be any complement of H0 in Φ(H). Then H/C is a covering group of Q and all covering groups Q˜ are obtained in this way as central quotients of H. Since there are 2 2 pn (n−1)/2 complements of H0 in Φ(H), we obtain pn (n−1)/2 such central quotients but, of course, different complements may give rise to isomorphic covering groups. In this paper we will distinguish a special class of covering groups and investigate the automorphism groups of the members of this class. 1 On the automorphisms of the group H The group H is a quotient of a free group by a fully invariant subgroup. As Higman observes, this means that H enjoys the following special property with respect to its automorphism group. Let M and N be subgroups of Φ(H) and suppose that there is an isomorphism between H/M and H/N. Then there exists an automorphism τ of H which satisfies τ(M) = N and induces the isomorphism between H/M and H/N. Conversely, it is clear that, given any subgroup M of Φ(H) and any automorphism σ of H, σ(M) is then also a subgroup of Φ(H) and σ induces an isomorphism between H/M and H/σ(M). This phenomenon allows us to relate the automorphisms of covering groups Q˜ to the group Aut(H) of all automorphisms of H. We first determine some information about Aut(H). In general if G is a group and σ ∈ Aut(G), we say that σ is a central automorphism of G if σ induces the identity on G/Z(G), or equivalently if σ fixes every coset of Z(G) in G. Lemma 1.1 The group Aut(H) contains a normal elementary abelian subgroup of order 2 pn (n+1)/2 consisting of central automorphisms and the quotient is isomorphic to GL(n, p) (the group of all automorphisms of Q). Proof: We note that H/Φ(H) =∼ Q and thus we have a homomorphism from Aut(H) to Aut(Q) = GL(n, p). Now the discussion above about lifting isomorphisms of quotients of H implies that the homomorphism considered above is surjective. We have therefore only to 2 determine the kernel of this automorphism. Suppose then that for some automorphism σ of H we have σ(x)Φ(H) = xΦ(H) for all x in H. It follows then that σ(x) = xτ(x), where τ(x) ∈ Φ(H). Thus σ is a central automorphism of H. It is straightforward to see that τ defines a homomorphism from H into Φ(H). Conversely, any such homomorphism from H into Φ(H) determines a central automorphism. Thus the number of central automorphisms is the number of homomorphisms from H into Φ(H). Now Φ(H) is an elementary abelian group of order pn(n+1)/2 and the largest elementary abelian quotient group of H is H/Φ(H) n n2(n+1)/2 of order p . It follows that there are p central automorphisms, as claimed. We now prove a theorem of P. Hall, [2], formula (18), p.202. While several proofs can be found in the literature, we include this proof, as it illustrates the utility of the group H, and also as we intend to extend the formula in the next section. Theorem 1.2 Let Γ1,..., Γr be representatives of all the non–isomorphic covering groups of Q. Then r 1 X 1 = |GL(n, p)| |Aut(Γ )| i=1 i 2 Proof: Aut(H) acts on the pn (n−1)/2 complements of H0 in Φ(H), and it follows from Higman’s argument that two complements are in the same orbit precisely when their corre- sponding central quotient groups are isomorphic covering groups of Q. Suppose for each i that ∼ Γi = H/Ci for some complement Ci and let Ai be the subgroup of Aut(H) that fixes Ci. We have then a homomorphism from Ai into Aut(Γi), and Higman’s argument implies that this homomorphism is surjective. Furthermore, the kernel of the homomorphism consists of those automorphisms of H that satisfy σ(x)Ci = xCi for all x ∈ H. It is easy to see that 2 there are pn such central automorphisms. The orbit–stabilizer theorem implies that r n2(n−1)/2 X p = |Aut(H): Ai| i=1 r X n2(n+1)/2 n2 = p |GL(n, p)|/p |Aut(Γi)|, i=1 and this gives us the required result. We note that Hall gives an illustration of this formula when considering the ten non– isomorphic covering groups of an elementary abelian group of order 8, [2], p. 203. 3 n2(n−1)/2 Each covering group Γi considered above has p central automorphisms, which form a normal elementary abelian subgroup of Aut(Γi). Thus in the formula of Theorem n2(n−1)/2 1.2, p divides each term |Aut(Γi)|. U. Webb shows in [6], Theorem 2, that for n2(n−1)/2 odd p, as n → ∞, the proportion of covering groups Γi for which |Aut(Γi)| = p approaches 1. Clearly, any attempt to classify such groups by conventional invariants such as automorphisms is unlikely to succeed. 2 The p–th power homomorphism Since H is nilpotent of class 2, we have (xy)p = xpyp[x, y]p(p−1)/2 for all x and y in H. Now H0 has exponent p and we deduce that (xy)p = xpyp if p is odd. We assume henceforth that p is odd. Thus x 7→ xp is a homomorphism and p p p the p–th powers of elements in H form a subgroup H , generated by h1,..., hn, which is a canonical complement to H0 in Φ(H). The corresponding quotient H/Hp is distinguished by the fact that it is the unique covering group Q˜ of exponent p. It is straightforward to see that this covering group admits a subgroup Γ, say, of automorphisms isomorphic to GL(n, p) and the full automorphism group is the semi–direct product of the group of central automorphisms with the subgroup Γ. Let now C be a complement of H0 in Φ(H) and let G = H/C be the corresponding covering group of Q.
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