Computation and Homotopical Applications of Induced Crossed Modules Ronald Brown Christopher D Wensley Mathematics Division School of Informatics University of Wales, Bangor Gwynedd, LL57 1UT U.K. email: {r.brown, c.d.wensley}@bangor.ac.uk September 6, 2002 Bangor Mathematics Preprint 02.04 Abstract We explain how the computation of induced crossed modules allows the co* *mpu- tation of certain homotopy 2-types and, in particular, second homotopy gro* *ups. We discuss various issues involved in computing induced crossed modules and g* *ive some examples and applications. Introduction The interactions between topology and combinatorial and computational group the* *ory are largely based on the fundamental group functor ß1 : (based spaces)! (groups). At the beginning of the 20th century there was an aim to to generalise the non * *commutative fundamental group to higher dimensions, hopes which seemed to be dashed in 1932* * by the proof that the definition of higher homotopy groups ßn then proposed by ~Ce* *ch led to commutative groups for n > 2. 1 Nonetheless, in the late 1930s and 1940s J.H.C. Whitehead developed properti* *es of the second relative homotopy group functor 2 : (based pairs of spaces)! (crossed modules), (X, A, a)7! (@ : ß2(X, A, a) ! ß1(A, a)) , where a 2 A X (see Section 4). Mac Lane and Whitehead showed in 1950 [24] th* *at crossed modules modelled homotopy 2-types (3-types in their notation) and evide* *nce has grown that crossed modules can be regarded as `2-dimensional groups'. Part of t* *his evidence is the 2-dimensional version of the Van Kampen Theorem proved by Brown and Higg* *ins in 1978 [9], which allows new computations of homotopy 2-types and so second ho* *motopy groups. This result should be seen as a higher dimensional, non commutative, l* *ocal-to- global theorem, illustrating themes in Atiyah's article [4]. It is interesting* * to note that the computation of these second homotopy groups is obtained through the computa* *tion of a larger non commutative structure. This work also throws emphasis on the p* *roblem of explicit computation with crossed modules, the discussion of which is the th* *eme of this paper. Our main emphasis in this paper is on induced crossed modules, which were de* *fined in [9] and studied further in papers by the authors [14, 15]. Given the crosse* *d module M = (~ : M ! P ) and a morphism of groups ' : P ! Q, the induced crossed module* * '*M has the form (@ : '*M ! Q), a crossed module over Q, and comes with a morphism * *of crossed modules ('*, ') : M ! '*M : '* // M _______'*M ~|| |@| fflffl| fflffl| P ____'__//Q . Their study requires a solution to many of the general computational problems o* *f crossed modules. In the case ~ = 0, when M is simply a P -module, '*M is the usual induced Q-* *module M ZP ZQ. Even in the case M = P, ~ = idP, we know of no relation between the induced * *crossed module (@ : '*P ! Q) and other standard algebraic constructions, although, inte* *restingly, im @ = NQ ('P ) the normal closure of 'P in Q. Thus the induced crossed module* * con- struction replaces this normal closure by a bigger group on which Q acts, and w* *hich has a universal property not usually enjoyed by NQ ('P ). A long-term project at Bangor is the development of a share library for the * *computa- tional group theory program GAP [18], providing functions to compute with these* * higher- dimensional structures. The first stage of this project saw the production of * *the library XMod1, containing functions for crossed modules and their derivations and for c* *at1-groups and their sections. The manual for XMod1 was included in [28] as Chapter 73. In* * particular, Alp [1] enumerated all isomorphism classes of cat1-structures on groups of orde* *r at most 47. This library has recently been rewritten for GAP4, with XMod2 included wit* *h the 4.3 2 release. Related libraries include Heyworth's IdRel [19] for computing identiti* *es among the relators of a finitely presented group, and Moore's GpdGraph and XRes [25] for * *computing with finite groupoids; group and groupoid graphs; and crossed resolutions. Thes* *e libraries are available at the HDDA website [20]. 1 Crossed modules A crossed module M (over P ) consists of a morphism of groups ~ : M ! P , calle* *d the boundary of M, together with an action of P on M, written (m, p) 7! mp, satisfy* *ing for all m, n 2 M, p 2 P the axioms: CM1) ~(mp) = p-1(~m)p , CM2) n~m = m-1nm . When CM1) is satisfied, but not CM2), the structure is a pre-crossed module [11* *, 21], having a Peiffer subgroup C generated by Peiffer commutators = m-1n-1m n~m , an* *d an associated crossed module (~0: M=C ! P ) with ~0induced by ~. Some standard algebraic examples of crossed modules are: (i)normal subgroup crossed modules (i : N ! P ) where i is an inclusion of a * *normal subgroup, and the action is given by conjugation; (ii)automorphism crossed modules (Ø : M ! Aut(M)) in which (Øm)(n) = m-1nm; (iii)abelian crossed modules (0 : M ! P ) where M is a P -module; (iv)central extension crossed modules (~ : M ! P ) where ~ is an epimorphism w* *ith kernel contained in the centre of M. For our purposes, an important standard construction is the free crossed Q-m* *odule F! = (@ : F (!) ! Q) on a function ! : ! Q, where is a set and Q is a group. The group F (!) ha* *s a presentation with generating set x Q and relators (m, q)-1 (n, p)-1 (m, q) (n, pq-1(!m)q) 8 m, n 2 , p, q 2 Q . The action is given by (m, q)p = (m, qp) and the boundary morphism is defined o* *n generators by @(m, q) = q-1(!m)q. This construction will be seen later as a special case o* *f an induced crossed module. The reader should be warned that the group F (!) can be very fa* *r from a free group: in fact, if ! maps all of to {1Q}, then F (!) is just the free Q-* *module on the set , and in particular is a commutative group. The major geometric example of a crossed module can be expressed in two ways* *. Let (X, A, a) be a based pair of spaces, with a 2 A X. The second relative homoto* *py group ß2(X, A, a) consists of homotopy classes rel J1 of continuous maps ff : (I2, `I2, J1) ! (X, A, a) 3 where I = [0, 1] and J1 = (I x {0, 1}) [ ({1} x I) I2. Each such ff is a map * *from the unit square I2 to the space X mapping three sides of the square to the point a and t* *he fourth side to a loop at a. Whitehead showed in [30] that there is a crossed module 2* *(X, A, a) with boundary map @ : ß2(X, A, a) ! ß1(A, a), ff 7! fi = ff(I x {0}) . The image of ff1 2 ß2(X, A, a) under the action of fi2 2 ß1(A, a) is illustrate* *d in the right- hand square of Figure 1. |___________________a||| |___________________a| |___________________a| | | | | | | | |@ | | | | | | a| | |@ fi2 fi2 | | | | | | | | | @I a ` | | | | | | |______|| | | @|_________| | | | | | | | | | | | | | |a a| |a a| |a a | |a a| | | | | | | | | ~ | | | | ~ |a | | a | | | | | | | | | | | | | |ff-1 | ff | ff | |ff-1 | ff | ff | | | | | | 2 | 1 | 2 | | 2 | 1 | 2 | | a| ff1 |a | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |____oe___-______-_|_|| |____oe__-_______-_|_|| |_____oe__-_______-_|_* *|| fi2 fi1 fi2 fi2 fi1 fi2 fi2 fi1 fi2 ff-12ff1ff2 ff@ff21 Figure 1: Verification of CM2) for 2(X, A, a) . Whitehead's main result in [29, 30, 31] was: Theorem 1.1 (Whitehead) If X is obtained from A by attaching 2-cells, then ß2* *(X, A, x) is isomorphic to the free crossed ß1(A, x)-module on the attaching maps of the * *2-cells. Later Quillen observed that if F ! E ! B is a based fibration, then the ind* *uced morphism of fundamental groups ß1F ! ß1E may be given the structure of a cross* *ed module. This fact is of importance in algebraic K-theory. We also note the following fact, shown in various texts on homological algeb* *ra or the cohomology of groups, e.g. [6], and which we relate to topology in section 4: 1.2 A crossed module M = (~ : M ! P ) determines algebraically a cohomology cla* *ss kM 2 H3(coker~, ker~), called the k-invariant of M, and all elements of this cohomology group have suc* *h a repre- sentation by a crossed module. 2 Other structures equivalent to crossed modules One aspect of the problem of higher dimensional group theory is that, whereas t* *here is essentially only one category of groups, there are at least five categories of * *equationally 4 defined algebraic structures which are equivalent to crossed modules, namely: o cat1-groups [23]; o group-groupoids [12]; o simplicial groups with Moore complex of length 1, [23]; o reduced simplicial T -complexes of rank 2, [16, 3, 26]; o reduced double groupoids with connection [13]. These categories have various geometric models. The 2-cells of some of these* * are illus- trated in the following pictures: e0 [ e1 [ e2 e0 [ e1 [ e2 2-simplex square ' $ ' $ - s |s______s_|- AA | | s s s A | | 6 ~ AK |6 |6 &% &%- s_______sA- |s______s_|||- crossed 2-groupoid, simplicial double module cat1-group T -complex groupoid There is also a polyhedral model, which allows rather general kinds of geometri* *c objects [22]. Thus, for computation in "2-dimensional group theory", decisions must be mad* *e as to which category to use to represent a given object, and to compute constructions* *. One reason for computing with the crossed module format is that this is closer to t* *he familiar realm of groups, for which many computational procedures and systems have been * *found and constructed. Part of the interest in computations with crossed modules is * *that such computations will also yield computations of these other structures, and this m* *akes them more familiar and understandable. 2.1 Cat1-groups In a cat1-group C = (e; t, h : G ! R) the embedding e : R ! G is a monomorphism* * while the tail and head homomorphisms t, h : G ! R are surjective and satisfy: CAT 1) te = he = idR, CAT 2) [kert, kerh] = {1G} . When CAT1) is satisfied, but nor CAT2), the structure is a pre-cat1-group with * *Peiffer subgroup [kert, kerh]. A cat1-group C determines a crossed module (@ : S ! R) * *where S = kert and @ = h|S. Conversely, a crossed module (~ : M ! P ) determines a ca* *t1-group (e; t, h : P n M ! P ) where t(p, m) = p and h(p, m) = p(~m). The axiom he = i* *dR is 5 equivalent to CM1) for a crossed module, while CAT2) is equivalent to CM2). Whe* *n ~ is the inclusion of the trivial subgroup in P , the associated cat1-group CP has e* * = t = h = idP. Note also that the semidirect product P n M admits a groupoid structure with* * t, h as source and target, and composition O where (p, m) O (p(~m), n) = (p, mn), makin* *g P n M a group-groupoid, i.e. a group internal to the category of groupoids. This no* *tion has a long history: the result that crossed modules are equivalent to group-groupoids* * goes back to Verdier, seems first to have been published in [12], and is used in [5]. Th* *e holomorph Aut (M) n M of a group M is the source of the cat1-group associated to the auto* *morphism crossed module (Ø : M ! Aut(M)). Now a colimit of cat1-groups colimi(ei; ti, hi: Gi! Ri) is easy to describe.* * One takes the colimits G0, R0of the underlying groups Gi, Ri, and finds that the endomorphism* *s ei, ti, hi induce endomorphisms e0: R0! G0and t0, h0: G0! R0satisfying axiom CAT1). The re- quired colimit is the cat1-group C00= (e00; t00, h00: G00! R0) which has G00= G* *0=[kert0, kerh0] and e00, t00, h00induced by e0, t0, h0. When C = (e; t, h : G ! R) and ' : R ! Q is an inclusion, the induced cat1-g* *roup '*C is obtained as the pushout of cat1-morphisms (e, idR) : CR ! C and (', ') : CR ! C* *Q . See Alp [1], [2] for further details. Further investigation is needed to see whether the use of cat1-groups can be* * shown to be more efficient than the direct method for the computation of some colimits o* *f crossed modules, particularly induced crossed modules,. The procedure has three stages:* * convert a crossed module M to a cat1-group C; calculate '*C; then convert '*C to '*M. 3 Computing colimits of crossed modules The homotopical reason for interest in computing colimits of crossed modules is* * the 2- dimensional Van Kampen Theorem (2-VKT) due to Brown and Higgins [9]. The formu- lation and proof of this theorem was found through the notion of double groupoi* *d with connection, since such structures yield an appropriate algebraic context in whi* *ch to handle both ä lgebraic inverses to subdivision", and the öh motopy addition lemma" (wh* *ich gives a formula for the boundary of a 3-cube). One form of the 2-VKT states that Whitehead's fundamental crossed module fun* *ctor 2 : (based pairs of spaces)! (crossed modules) preserves certain colimits. So for the calculation of certain homotopy invarian* *ts, we need to know how to calculate colimits of crossed modules. To this end, we start by * *using some elementary category theory. The forgetful functor (crossed modules) ! (groups), (~ : M ! P ) 7! P , has * *a right adjoint P 7! (i : P ! P ), and so preserves colimits. This shows how to comput* *e the 1-dimensional part of the colimit crossed module in terms of colimits of groups. The aim now is to transfer the problem to computing colimits of crossed modu* *les over a fixed group P . To do this, suppose given a morphism of groups ' : P ! Q. The* *n there is 6 a pullback functor '* : (crossed modules over Q)! (crossed modules over P.) This functor has a left adjoint '* : (crossed modules over P!)(crossed modules over Q), which gives our induced crossed module. This construction can be described as a* * "change of base" [8]. To compute a colimit colimi(~i: Mi! Pi), one forms the group P = * *colimiPi, and uses the canonical morphisms OEi : Pi ! P to form the family of induced cro* *ssed P - modules ((~i)* : (OEi)*Mi! P ). The colimit of these in the category of crossed* * P -modules is isomorphic to the original colimit. Now if M0 is the colimit in the category* * of groups of the (OEi)*Mi, then there is a canonical morphism M0 ! P and an action of P on M* *0. The resulting (M0 ! P ) is a pre-crossed module, and quotienting by its Peiffer sub* *group gives the required crossed module. Presentations for induced crossed modules were given in [9], and more recent* *ly fami- lies of explicit examples have been computed, partly by hand and partly using G* *AP [14]. Computation of induced crossed modules is here reduced to problems of computati* *on in combinatorial group theory. A key fact which makes one expect successful comput* *ations is that if (~ : M ! P ) is a crossed module with M finite, and if ' : P ! Q is a m* *orphism of finite index, then the induced crossed Q-module '*M is also finite [14, Theorem* * 2.1]. Example 3.1 When ~ : M ! P and ' : P ! Q are subgroup inclusions, there are c* *omplete descriptions of '*M in the following cases: (i)If ' is surjective then '*M ~=M=[M, ker'], ([9, Proposition 9]). (ii)If M is abelian and '~(M) is normal in Q then '*M is abelian and is the us* *ual induced Q-module M ZP ZQ, ([14, Corollary 1.6]). (iii)If M and P are normal subgroups of Q then '*M ~=M x (Mab I(Q=P )), whe* *re I denotes the augmentation ideal. If in addition M = P then '*P ~=P x (P ab* *)[Q:P]-1, ([15, Theorem 1.1]). (iv)If M = P = C2, the cyclic group of order 2, ~ = idP, and ' : C2 ! D2nis th* *e inclusion to a reflection in the dihedral group D2n, then '*P ~= D2n ([14, Example 1* *.4]). The action is not the usual conjugation: when n is odd the boundary is an isom* *orphism, but when n is even the kernel and cokernel are isomorphic to C2. 4 Homotopical applications As explained in the introduction, the fundamental crossed module functor 2 ass* *igns a crossed module (@ : ß2(X, A, a) ! ß1(A, a)) to any based pair of spaces (X, A, * *a). Theorem C of [9] is a 2-dimensional Van Kampen type theorem for this functor. We will * *use the following consequence: 7 Theorem 4.1 ([9], Theorem D) Let (B, V, b) be a cofibred pair of spaces, let * *f : V ! A be a based map, and let X be the pushout A [f B in the left-hand diagram below. Su* *ppose also that A, B, V are path-connected, and (B, V, b) is 1-connected. Then the based p* *air (X, A, a) is 1-connected and the right-hand diagram f ~* V ____//_A ß2(B, V, b)____//_ß2(X, A, a) || || ffi|| ffi0|| fflffl|fflffl| fflffl| fflffl| B ____//_X ß1(V, b)________//ß (A, a) ~ 1 presents ß2(X, A, a) as the crossed ß1(A, a)-module ~*(ß2(B, V, b)) induced fro* *m the crossed ß1(V, b)-module ß2(B, V, b) by the group morphism ~ : ß1(V, b) ! ß1(A, a) induc* *ed by f. As pointed out earlier, when P is a free group on a set and ~ is the ident* *ity, the induced crossed module '*P is the free crossed Q-module on the function '| : * * ! Q. Thus Theorem 4.1 implies Whitehead's Theorem as stated in Theorem 1.1. A consid* *erable amount of work has been done on this case, because of the connections with iden* *tities among relations, and methods such as transversality theory and "pictures" have proved* * successful ([11, 27]), particularly in the homotopy theory of 2-dimensional complexes [21]* *. However, the only route so far available to the wider geometric applications of induced * *crossed modules is Theorem 4.1. We also note that this Theorem includes the relative Hurewicz T* *heorem in this dimension, on putting A = V , and f : V ! V the inclusion. We will apply this Theorem 4.1 to the classifying space of a crossed module,* * as defined by Loday in [23] or Brown and Higgins in [10]. This classifying space is a functor* * B assigning to a crossed module M = (~ : M ! P ) a based CW -space BM with the following prope* *rties: 4.2 The homotopy groups of the classifying space of the crossed module M = (~ :* * M ! P ) are given by 8 < coker~ for i = 1, ßi(BM) ~= ker~ for i = 2, : 0 for i > 2 . The first Postnikov invariant of BM is precisely the k-invariant of M as in 1.2. 4.3 The classifying space BP = B(i : 1 ! P ) is the usual classifying space of * *the group P , and BP is a subcomplex of BM. Further, there is a natural isomorphism of c* *rossed modules 2(BM, BP, x) ~= M . 4.4 If X is a reduced CW -complex with 1-skeleton X1, then there is a map X ! B( 2(X, X1, x)) inducing an isomorphism of ß1 and ß2. 8 It is in these senses that it is reasonable to say, as in the Introduction, * *that crossed modules model all based homotopy 2-types. We now give two direct applications of Theorem 4.1. Corollary 4.5 Let M = (~ : M ! P ) be a crossed module, and let ' : P ! Q be a morphism of groups. Let fi : BP ! BM be the inclusion. Consider the pushout fi BP _______//BM . B'|| || fflffl| fflffl| BQ ___fi0__//X Then the fundamental crossed module of the pair (X, BQ, x) is isomorphic to the* * induced crossed module (@ : '*M ! Q), and this crossed module determines the 2-type of * *X. In particular, the second homotopy group ß2(X, x) is isomorphic to ker@. Proof The first statement is immediate from Theorem 4.1. The second statement * *follows from results of [10], since the morphism Q ! ß1(X) is surjective. The final st* *atement follows from the homotopy exact sequence of the pair (X, BQ, x). * * 2 Remark 4.6 An interesting special case of the last Corollary is when M is an * *inclusion of a normal subgroup, since then BM is of the homotopy type of B(P=M). So we h* *ave determined the 2-type of a homotopy pushout Bp BP _____//BR B'|| || fflffl| fflffl| BQ _p0_//_X in which p : P ! R is surjective. * * 2 Corollary 4.7 Let ' : P ! Q be a morphism of groups, and let BP denote the co* *ne on BP . Then the fundamental crossed module 2(BQ [B' BP, BQ, x) is isomorphi* *c to the induced crossed module (@ : '*P ! Q). In particular, the second homotopy * *group ß2(BQ [B' BP, x) is isomorphic to ker@. We also note that in determining the crossed module representing a 2-type we* * are also determining the first Postnikov invariant of that 2-type. However it may be mor* *e difficult to describe this invariant as a cohomology class, though this is done in some case* *s in [14, 15]. 5 Computational issues Recall from Proposition 9 of [9] that when ' : P ! Q is a surjection then '*M ~* *=M=[M, K], where K = ker' and [M, K] denotes the subgroup of M generated by the m-1mk for * *all 9 m 2 M, k 2 K. When ' is neither surjective nor injective, we obtain a factorisa* *tion ' = '2O'1 with '1 surjective and '2 injective, and construct the induced crossed module i* *n two stages: ('1)* ('2)* M ______//_('1)*M____//_('2)*('1)*M ~|| @1|| @2|| fflffl| fflffl| fflffl| P ____'1__//im'____'2_____//_Q . The first stage is easily constructed as a quotient group, so in the following * *subsections we discuss significant computational issues in the case when both ' and ~ are s* *ubgroup inclusions. Note that computation of free crossed modules, as described in section 1, is* * in general difficult since the groups are usually infinite, and is not attempted in the cu* *rrent version of the package. 5.1 Isomorphic pairs of groups The GAP function IsomorphismPermGroup enables the construction of a permutation* * group isomorphic to a finite group already obtained. Thus it is sufficient to impleme* *nt the induced construction for permutation groups. Similarly, the function IsomorphismFpGroup* * enables the construction of finitely presented groups F M, F P, F Q isomorphic to permu* *tation groups M, P, Q; monomorphisms F ~ : F M ! F P, F ' : F P ! F Q mimicing the inclusi* *ons M ! P ! Q; and an action of F P on F M. 5.2 Copower of groups The construction of induced crossed modules, described in [9, 14], involves the* * copower F M ~*T , namely the free product of groups F Mt, t 2 T , each isomorphic to F * *M. Here T is a transversal for the right cosets of F P in F Q, in which the representat* *ive of the subgroup F P is taken to be the empty word. The group F Mt = {(m, t) | m 2 F M} has product (m, t)(n, t) = (mn, t) and F Q acts by (m, t)q = (mp, u) where tq =* * ('p)u in F Q. If F M has fl generators, then a presentation F C of M ~*T with fl | T | g* *enerators may be constructed using functions in the GAP Tietze package ([18], Chapter 46* *). The relators of F C comprise | T | copies of the relators of F M, suitably renumber* *ed. Define ffi0: F C ! F Q, (m, t) 7! t-1('~m)t. 5.3 Quotient by the Peiffer subgroup Let be a generating set for F M and let FP be the closure of under the act* *ion of F P . Then '*(M) ~=F C=F N where F N is the normal closure in F C of the Peiffer elem* *ents 0(n,s) FP <(n, s), (m, t)> = (n, s)-1(m, t)-1(n, s)(m, t)ffi (m, n 2 , s,* *(t12)T ). 10 The homomorphism '* is induced by the projection pr1m = (m, 1FQ ) onto the firs* *t factor, and the boundary ffi of '*M is induced from ffi0as shown in the following diagr* *am: '* // F M _______F C=F N . ~|| |ffi| fflffl| fflffl| F P___'_____//F Q Thus a finitely presented group F I ~= '*M is obtained by adding to the relator* *s of F C further relators corresponding to the list of elements in equation (1), and the* * presentation may be simplified by applying Tietze transformations. 5.4 Tracing Tietze transformations As well as returning an induced crossed module, the construction should return * *a morphism of crossed modules ('*, ') : M ! '*M. When Tietze transformations are then appl* *ied to the initial presentation for F I, during the resulting simplification some of the f* *irst fl generators may be eliminated, so the projection pr()may be lost. In order to preserve this* * projection, and so obtain the morphism '*, it is necessary to record for each eliminated ge* *nerator g a relator gw-1 where w is the word in the remaining generators by which g was eli* *minated. A significant advantage of GAP is the free availability of the library code,* * which enables the user to modify a function so as to return additional information. For the X* *Mod1 version of the package, the Tietze transformation code was modified so that the resulti* *ng presenta- tion contained an additional field presI.remember, namely a list of (at least) * *fl | T | relators expressing the original generators in terms of the final ones. In more recent r* *eleases of GAP an equivalent facility has been made generally available using the TzInitGenera* *torImages function. 5.5 Polycyclic groups Recall that a polycyclic group is a group G with power-conjugate presentation h* *aving gen- erating set {g1, . .,.gn} and relations {goii= wii(gi+1, . .,.gn), ggji= w0ij(gj+1, . .,.gn) 8 1 6 j < i 6* *(n}.2) (see [18], Chapters 43,44). These are implemented in GAP as PcGroups. Since sub* *groups M 6 P 6 G have induced power-conjugate presentations, if T is a transversal for* * the right cosets of P in G, then the relators of M ~*T are all of the form in (2). Furth* *ermore, all the Peiffer relations in equation (1) are of the form ggji= gpk, so one might h* *ope that a power conjugate presentation would result. Consideration of the cyclic-by-cycli* *c case in the following example shows that this does not happen in general. Example 5.1 Let Cn be cyclic of order n with generator x, and let ff : x 7! x* *a be an automorphism of Cn of order p. Take G = ~= Cp n Cn* *. When M = P = Cn . G cases (ii) and (iii) of example 3.1 apply, and '*Cn ~=Cpn. 11 It follows from the relators that hig = ghai, 0 < i < n, and that h-1(ghi* *(1-a))h = gh(i+1)(1-a). So if we put gi= ghi(1-a), 0 6 i < n, then ggji= g[j+a(i-j)]. Wh* *en M = P = Cp = and ' : Cp ! G, we may choose as transversal T = {1G, h, h2, . .,* *.hn-1}. Then M ~*T has generators {(g, hi) | 0 6 i < n}, all of order p, and relators {(g, h* *i)p | 0 6 i < n}. The additional Peiffer relators in equation (1) have the form (g, hi)(g, hj) = (g, hj)(gk, hl) where hih-jghj = gkhl so k = 1 and l = [j + a(i - j)]. Hence ` : '*M ! Q, (g, hi) 7! gi is an isomo* *rphism, and '*M is isomorphic to the identity crossed module on Q. Furthermore, if we * *take M to be a cyclic subgroup Cm of Cp then '*M is the normal subgroup crossed mod* *ule (i : Cm n Cn ! Cp n Cn). * * 2 5.6 Identifying '*M From some of the special cases listed in example 3.1 and from other examples, w* *e know that many of the induced groups '*M are direct products. However the generating* * sets in the presentations that arise following the Tietze transformation do not in gene* *ral split into generating sets for direct summands. This is clearly seen in the following simp* *le illustration. Example 5.2 Let Q = S4, the symmetric group of degree 4, and M = P = A4, the alternating subgroup of Q of index 2. Since the abelianisation of A4 is cyclic * *of order 3, case (iii) in section 3 shows that '*M ~= A4 x C3. However a typical presentation fo* *r A4 x C3 obtained from the program is , and one generator for the C3 summand is yzx2. Converting to a permutation group* * H gives a degree 12 representation with generating set {(2, 9, 4)(3, 5, 6)(8, 12, 10), (1, 4, 2)(3, 5, 7)(10, 11, 12), (1, 8, 3)(* *2, 10, 5)(7, 9, 12)}. Converting H to an PcGroup produces a 4-generator group with subnormal series I . C2 . C22. A4 . A4 x C3 , where each extension adds a generator gi, i = 1 . .4.and g1g2g4 is a generator * *for the normal C3. In these representations, the cyclic summand remains hidden, and an explic* *it search among the normal subgroups must be undertaken to find it. * * 2 6 Results In this section we list the crossed modules induced from subgroups of groups of* * order at most 23 (excluding 16), except that the special cases mentioned earlier enable * *us to exclude 12 abelian and dihedral groups; the case when P is normal in Q; and the case when * *Q is a semidirect product Cm n Cn. In the first table, we assume given an inclusion ' : P ! Q of a subgroup P o* *f a group Q, and a normal subgroup M of P . We list the isomorphism type of the source of th* *e crossed module (@ : '*M ! Q) induced from (~ : M ! P ) by the inclusion '. Recall that * *this kernel is realised as a second homotopy group in corollary 4.5. Labels I, Cn, D2n, An* *, Sn denote the identity, cyclic, dihedral, alternating and symmetric groups of order 1, n,* * 2n, n!=2 and n! respectively. The group Hn is the holomorph of Cn and H+nis its positive sub* *group in degree n. SL(2, 3) and GL(2, 3) are the special and general linear groups of o* *rder 24, 48 respectively. Labels of the form m.n refer to the nth group of order m accordi* *ng to the GAP4 numbering. ________________________________________ | | |Q|M| |P | Q | ' M | ker@ | | |_|_____|_|___|________|___*____|______|_|_ | | 12C ||C | A | H+ | C | | | | 2|| 2 | 4 | 8 | 4 | | | | C || C | A | SL(2, 3) |C | | | | 3|| 3 | 4 | | 2 | | | | 18C ||C |C n C2 | 54.8 | C | | | | 2|| 2 | 2 3 | | 3 | | | | S| |S |C n C2 | 54.8 | C | | | | |3| 3 | 2 3 | | 3 | | | | 20C ||C | H | D | C | | | | 2|| 2 | 5 | 10 | 2 | | | | C || C2 | D | D | I | | | | 2|| 2 | 20 | 10 | | | | | C2|| C2 | D | D | I | | | | 2|| 2 | 20 | 20 | | | | | 21C ||C | H+ | H + | I | | |_|_____3||_3_|___7____|___7____|______|_| Table 1 The second table contains the results of calculations with Q = S4, where C2 * *= <(1, 2)>, C02= <(1, 2)(3, 4)>, and C22= <(1, 2), (3, 4)>. The final column contains the a* *utomorphism group Aut('*M) (where known). ___________________________________________ | | M |P | ' M |ker@ |Aut(' M) | | |_|____|_____|____*______|______|_____*___|_|_ | | C |C | GL(2, 3) |C | S C | | | | 2 | 2 | | 2 | 4 2 | | | | C |C |C SL(2, 3) |C | 144.183 | | | | 3 | 3 | 3 | 6 | | | | | C |S | SL(2, 3) |C | S | | | | 3 | 3 | | 2 | 4 | | | | S |S | GL(2, 3) |C | S C | | | | 3 | 3 | | 2 | 4 2 | | | | C0 |C0 | 128.? |C C3 | | | | | 2 | 2 | | 4 2 | | | | | C0C|2, C | H+ | C | S C | | | | 2 |2 4 | 8 | 4 | 4 2 | | | | C0 |D | C3 | C | SL(3, 2) | | | | 2 | 8 | 2 | 2 | | | | | C2 |C2 | S C | C | S C | | | | 2 | 2 | 4 2 | 2 | 4 2 | | | | C2 |D | S | I | S | | | | 2 | 8 | 4 | | 4 | | | | C |C | 96.219 |C | 96.227 | | | | 4 | 4 | | 4 | | | | | C |D | S | I | S | | | | 4 | 8 | 4 | | 4 | | | | D |D | S C | C | S C | | |_|__8_|_8___|____4_2____|__2__|____4_2___|_| Table 1 13 An interesting problem is to obtain a clearer understanding of the geometric* * significance of these tables. 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