2 LOCAL CONJUGACY IN GL2(Z=p Z) SPUR FINAL PAPER, SUMMER 2016 HYUN JONG KIM MENTOR: ATTICUS CHRISTENSEN PROJECT SUGGESTED BY ANDREW SUTHERLAND Abstract. Subgroups H1 and H2 of a group G are said to be locally conjugate if there is a bijection f : H1 ! H2 such that h and f(h) are conjugate in G. We study local conjugacy 2 among subgroups of GL2(Z=p Z), where p is an odd prime, building on Andrew Sutherland's categorizations of subgroups of GL2(Z=pZ) and local conjugacy among them. We obtain 2 a classification of locally conjugate subgroups of GL2(Z=p Z) in the kernel of the natural 2 map ' : GL2(Z=p Z) ! GL2(Z=pZ). We further inspect local conjugacy among subgroups 2 of GL2(Z=p Z) using this classification. Contents 1. Introduction 2 2. Notation 2 2.1. Groups 2 3. Properties of Locally Conjugate Subgroups 3 2 4. Subgroups of the Kernel of the Homomorphism GL2(Z=p Z) ! GL2(Z=pZ) 4 4.1. Orbits of Mat2(Z=pZ) under conjugation by elements of GL2(Z=pZ) 4 2 4.2. Subgroups in a Subgroup of ker ' Up to Conjugacy by GL2(Z=p Z) 5 2 4.3. Subgroups of ker ' Up to Conjugacy by GL2(Z=p Z) 7 2 5. Locally Conjugate Subgroups of GL2(Z=p Z) 11 5.1. Subgroups of GL2(Z=pZ) 11 5.2. Stabilizers of Subgroups of ker ' 13 5.3. Applying the Schur-Zassenhaus Theorem 15 2 5.4. Representations of Elements of GL2(Z=p Z) 15 5.5. Conjugacy and Local Conjugacy 16 Acknowledgement 22 References 23 1 1. Introduction Sutherland was led to consider subgroups of GL2(Z=pZ) which are locally conjugate but not necessarily conjugate as he was developing an algorithm to compute the images of Galois representation given by the Galois action on p-torsion points of elliptic curves. In his paper, Sutherland fully identifies nontrivially locally conjugate subgroups of GL2(Z=pZ) (see Theo- k rem 2), where p is an odd prime. Our hope is to understand local conjugacy in GL2(Z=p Z) where k ≥ 2 and also in the projective limit GL2(Zp). In this paper, we work with the case 2 when k = 2. We eventually eliminate some pairs of subgroups of GL2(Z=p Z) that cannot be nontrivially locally conjugate (see Corollary 3, Remark 8). Sutherland's result with k = 1 and our observations with k = 2 and p = 3 lead us to believe that locally conjugate subgroups k of GL2(Z=p Z) come in pairs up to conjugacy and that the subgroups are isomorphic. 2. Notation Throughout this paper, p is an odd prime. ϵ is taken to be a nonsquare of Z=pZ. For a group G Sand g 2 G, we take gG to be the conjugacy class of g in G. Furthermore, ⊆ G G for S G, S = s2S s . Definition 1. Given 2×2 matrices M1 and M2, we say that they are diagonally swapped or that they are diagonal swaps if M2 is obtained by switching the diagonal entries of M1. Moreover, given groups of matrices H1 and H2, we say that they are diagonally swapped if the elements of H2 are diagonal swaps of elements of H1. 2.1. Groups. GL2(R); SL2(R) and PGL2(R) denote the general, special and projective lin- ear groups of 2 × 2 matrices over a ring R. In particular, we define the following subgroups of GL2(Z=pZ): Definition 2. {( ) } w 0 Z(p) = 2 GL (Z=p2Z) 0 w 2 {( ) } w 0 C (p) = 2 GL (Z=p2Z) s 0 z 2 {( ) } w ϵy C (p) = 2 GL (Z=p2Z) ns y w 2 {( ) } w x B(p) = 2 GL (Z=p2Z) ; 0 z 2 are called the center, Cartan-split subgroup, Cartan-nonsplit subgroup and Borel subgroup of GL2(Z=pZ) respectively. For H ≤ GL2(Z=pZ), let N(H) denote the normalizer of H in GL2(Z=pZ). In particular, ( ) 0 1 N(C (p)) = C (p) [ C (p) s s 1 0 s ( ) 1 0 N(C (p)) = C (p) [ C (p): ns ns 0 −1 ns 2 3. Properties of Locally Conjugate Subgroups In this section, we define locally conjugate subgroups and discuss some properties of local conjugacy. Definition 3. Let G be a group and H1;H2 ≤ G. (G; H1;H2) is a Gassman triple if there is a bijection f : H1 ! H2 such that h and f(h) are conjugate in G for all h 2 H1. We also say that H1 and H2 are locally conjugate in G. If H1 and H2 are conjugate in G, then they are also locally conjugate in G, in which case we say that they are trivially locally conjugate. Note that local conjugacy between subgroups H1 and H2 of G is determined not only by H1 and H2, but also by G. For example, we will see later that the kernel of the natural 2 homomorphism ' : GL2(Z=p Z) ! GL2(Z=pZ) has subgroups that are nontrivially locally 2 conjugate with respect to GL2(Z=p Z). However, such subgroups are not locally conjugate with respect to ker ' because ker ' is abelian, in which case locally conjugate subgroups are always equal. Proposition 1. Let G be a group and H1;H2 ≤ G. H1 and H2 are locally conjugate in G if and only if jH1 \ Cj = jH2 \ Cj for all conjugacy classes C of G. Proof. Suppose H1 and H2 are locally conjugate in G via f : H1 ! H2. For every conjugacy j \ −1 j \ class C of G, f H1\C maps into H2 C. Likewise, f H2\C maps into H1 C and is the j j \ j j \ j inverse of f H1\C . Thus, H1 C = H2 C . Conversely, suppose jH1 \ Cj = jH2 \ Cj for every conjugacy class C of G. Choose some bijections fC : H1 \ C ! H2 \ C and define f : H1 ! H2 as f(h) = fhG (h). f is a well defined bijection because the conjugacy classes of G partition G. Moreover, h and f(h) are in the same conjugacy class for every h 2 H1, so H1 and H2 are locally conjugate. Proposition 2. Let G be a group, H1;H2 ≤ G and N C G. If H1 and H2 are locally conjugate in G, then H1 \ N and H2 \ N are locally conjugate in G. Proof. N is the disjoint union of some conjugacy classes of G. Let C be a conjugacy class of G. If C ⊆ N, then j(Hi \ N) \ Cj = jHi \ Cj for i = 1; 2. Otherwise, j(Hi \ N) \ Cj = 0. H1 \ N and H2 \ N are therefore locally conjugate in G by Proposition 1 0 0 Proposition 3. Let G; G be finite groups, H1;H2 ≤ G and ' : G ! G a surjective homomorphism. If H1 and H2 are locally conjugate in G, then '(H1) and '(H2) are locally conjugate in G0. S 0 0 −1 G Proof. Let C be any conjugacy class of G and let U = x2C0 (' (x)) . We claim that '−1(C0) = U. If d 2 '−1(C0), then '(d) 2 C0, in which case '(d) 2 ('−1('(d))G ⊆ U. Therefore, '−1(C0) ⊆ U. Conversely, if d 2 ('−1(x))G for some x 2 C0, then d = gyg−1 for some g 2 G and y 2 '−1(x). It follows that '(d) = '(g)'(y)'(g)−1 = '(g)x'(g)−1, and so d 2 '−1(C0). Hence, U ⊆ '−1(C0), as desired. In particular, '−1(C0) is the union of conjugacy classes of G. ker ' is the union of conjugacy classes of G because it is normal in G. Moreover, since H1 −1 0 and H2 are locally conjugate, jH1 \ ker 'j = jH2 \ ker 'j. Similarly, jH1 \ ' (C )j = jH2 \ −1 0 0 −1 0 0 −1 0 ' (C )j. Note that '(Hi)\C = '(Hi\' (C )), so j'(Hi\C )j = jHi\' (C )j=jHi\ker 'j 0 0 for i = 1; 2. Thus, j'(H1) \ C j = j'(H2) \ C j. 3 Lemma 1. Let G be a finite group and H1;H2 ≤ G with H1 and H2 locally conjugate in G. If H1 is cyclic, then so is H2 and H1 and H2 are conjugate. Proof. Say that h1 generates H1. There is some h2 2 H2 that is conjugate to h1. h1 and h2 have the same order, which is jH1j = jH2j, so h2 generates H2. 2 4. Subgroups of the Kernel of the Homomorphism GL2(Z=p Z) ! GL2(Z=pZ) 2 For the rest of the paper, ' denotes the natural homomorphism GL2(Z=p Z) ! GL2(Z=pZ) unless stated otherwise. Elements of ker ' are of the form I + Ap, where we may identify A 4 as an element of Mat2(Z=pZ). ker ' ' (Z=pZ) because (I +A1p)(I +A2p) = I +(A1 +A2)p. Therefore, ker ' is a Z=pZ vector space of dimension 4. 2 By Propositions 2 and 3, H1 \ ker ' and H2 \ ker ' are locally conjugate in GL2(Z=p Z) and '(H1) and '(H2) are locally conjugate in GL2(Z=pZ) if H1 and H2 are locally conjugate 2 in GL2(Z=p Z). Sutherland fully identifies local conjugacy in GL2(Z=pZ) (see Theorem 2). In this section, we determine local conjugacy between subgroups of ker ' with respect to 2 GL2(Z=p Z) up to conjugacy. To categorize the subgroups of ker ' up to conjugacy, we conjugate the subgroups to pick some generators to be of desired form.