ON GENERAL GROUP EXTENSION by John D•. Dixon, M.A. A thesis submitted to the Faculty of Graduate Studies and Research in partia1_ fulfilment of the requirements for the degree of Doctor of Philosophy. Department of Mathematics, McGill University, Montreal. April 196~. 2 TABLE OF CONTEN~S Page Introduction 3 1. Mixed Groups 6 2. Extension Problem 12 3. Cardinal Representations of Mp(l) 21 4. Extension Functions 27 5. Particular Extensions 35 6. Structure of Normal Extensions 44 7. Structure of General Extensions 55 Notes 60 Bibliography 63 INTRODUCTION The problem of normal extensions in groups (see Kurosh [1] Chapt er XII) has be en studied in consider­ able detail and has reached the point of a reasonable solution. The corresponding probi.em of general exten­ sions in groups, although studied quite extensively, has not yet reached that point. The aim of the pres­ ent thesis is to generalise some of the resulte ob­ tained in the c.ase of normal extensions to the general case. It was proved by Baer t11 that the cosets of a group H modulo an arbitrary subgroup G can be given a structure called a mixed group. In partic­ ular when G is normal in H the mixed group is a group. Taking this generalisation of a quotient group as a mixed group as a starting point, this thesis de­ velops a generalisation of normal extension theory. (cf. the alternative generalisation of Szep [~ ). Whilst no generalisation of the cohomology theory in normal extensions is considered, the structure of the different extensions of a group in the general case is examined and in particular criteria for auch an exten­ sion to be possible are found. The NOTES at the end of the thesis give detailed references and specify the parts of the thesis claimed to be new resulta. The author would like to express his considerable gratitude to Professor H. Schwerdtfeger who eupervised this work., Prof. Schwerdtfeger suggested the original problem of extending groups by mixed groups and he has continued to give generous help and patient encourage­ ment in the execution of the thesis. Notation The following notation is used consistently: 1) (x : * 1 denotes the set of elements x which satisfy the given condition * • If A, B are two sets then AB = {ab : a E A, b E B} is defined when the products ab (a é A, b é B) are defined., In particular, if U is a subset of a group G then GU = G and u2 = U implies that U is a group when U is finite. 2) For a group H G ~ H means G is a eubgroup, G 4 H means that G is a normal subgroup of H • The unit group is written (1) • Z(H) denotes the centre of H and N(G;H) denotes the normaliser of G in H • The order of G and the index of G in H are denoted by (G:l) and (H:G) respectively. More gen­ erally (M:l) denotes the order of a set M • A x B is the direct product of the groups A and B • Sm will be used to denote the symmetric group on m let­ ters. We may suppose Sm-l ~ Sm for m = 2, 3, •• 3) If ~ is an arbitrary mapping of a set A into a set B then a~ (a ~ A) denotes the image of a in B • If C ç A then ~ \ C denotes the mapping ~ restricted to the elements of C • If ~ is a homomorphism between the groups H and H then ker ~ denotes the iernel of the homomorphism. (Bee Kurosh l~ _s 10). The sign # denotes the end of a proof. 6 1. MIXED GROUPS Following Baer [~ we define a mixed group ("Mischgruppe") as follows. Definition 1.1 A mixed group is a set M for which for certain ordered pairs of elements x,y G M a prod­ uct xy e M is defined such that (i) there is a subset U of M which under the given product is a group called the nucleus ("Kern") of M . ' (ii).* x y is defined if and only if x e U ; in particul ar for the unit element 1 é u ly = y ; (iii) products are associative when defined, i.e. x(yz) = (xy)z when x,y E U • *More exactly we are defining a right mixed group. A left mixed group could be defined similarly by alter­ ing (ii) and (iii) auch that xy is defined if and only if y é U • There is an obvious one-one correspondance between right and left mixed groups and in what follows we consider exclus ively the right mixed groups defined above. In particular every group is a mixed group whose nucleus is the whole set. On the other hand any set can be made a mixed group by introducing a left unit 1 element and defining the product only for left mult­ iplication by this unit. The nucleus of this mixed group is the unit group. Theorem 1.1 A mixed group M with nucleus U may be written as a union of disjoint cosets Ux (x E M). We write (1.1) M = U + Ua + Ub • ••• where the sets U = U.l, Ua, Ub, •• are disjoint. Proof Each element x E M lies in one coset, viz. Ux • It is therefore sufficient to prove that any two cosets Ux, Uy are either disjoint or Ux = Uy • Suppose Ux ~ Uy ~ ~ then there is a z ~ Ux A Uy and z = ux = vy for some u,v e.. U • But then Uz = U(ux) =(Uu)x = Ux becauee of the group property of U. Similarly Uz = Uy and so the theorem is proved. # Definition 1.2 The (finite or infinite) cardinal number of disjoint cosets (modulo U ) in M given in the decomposition (1.1) is called the~ of M • If M is finite with rank r we note (M:l) = r(U:l) • In particular the order of the nucleus div­ ides the order of the mixed group. Definition 1.3 Two mixed groups M1 and M2 are isomorphic (written M1 ~ M2 ) if there is a mapping 8 -c ( called an isomorphism) auch that ( i) -c is one-one from the who le of M1 onto M2 (ii) the product (a~)(b~) is defined in M2 if and only if ab is defined in M1 and then (a~)(b~) = (ab) -c • Definition 1.4 A set R containing exactly one elem- ent of M from each coset of M (mod U ) is called a set of representatives of M • Theorem 1.2 Two mixed groups Mi with nuclei u1 and ranke ri (i = 1,2) are isomorphic if and only if the groups ul and u2 are (group) isomorphic and rl = r2 • Pro of A. Let ""Cl be an isomorphism "t"1 : ul onto u2 and let rl = r2 • Let Rl and R2 be sets of repres- entatives for Ml and M2 respectively. Be cause rl = r 2 the re is a nne-one mapping T~ : R1 onto R2 • Each element x1 ~ Mi is in just one coset and so can be written uniquely xi = uiri (ui ~ Ui' ri 6 Ri) 1,2) • -c )-c (i = We define by (u1r 1 = (u1 L:,)(r1 -r.J (u1-c, 6 u2, r 1 ""C",_ ~ R2) • -c satisfi&S (i) of Definit­ ion 1. 3 a ince -c, and ""'C~ are each one-one. On the other band (x~)(y~) is defined in M2 if and only if X""t:. E:. U and hence X é Ul, X"C X 1: • Wri ting 2 = 1 y= ur (u é u1 , r é R1 ) we then have (x~)(y~) = (x -c, )( u ~. )( r ,;..) = (( xu) -c, )( r ~) ={.( xu) r 1-r: = {xy 3- -c • So (ii) of Definition 1.3 is satisfied and ~ is the required isomorphism between the mixed groups M1 and )(2 • B. Conversely, if we have an isomorphism ~ : M1 onto M2 then ~ju1 : u1 onto u2 is an isomorphism (of groups) between the nuclei of the mixed groups. ~ also gives a one-one mapping between the sets of cosets of the respective mixed groups and bence r 1 = r 2 • # From the above theorem a mixed group is given uniquely up to an isomorphism by prescribing its nucleus U and rank r • We shall denote this mixed group by Mr(U) • The next theorem shows the connexion between the concept of mixed group and the problem of group exten- sion. Theorem 1.3 Let H be a group with subgroup G 'H • We consider the set M of all cosets of H (mod G ) and define a product between cert~in ordered pairs of cosets by Gx oGy = Gz (x, y, z ~ H) if and only if the set product GxGy is a single coset Gz • M is a mixed group with nucleus U = {Gx : x é N(G;H)} where N(Q;H) is the normaliser of G in H • Proof U is a group with unit element G since for all x,y ~ N(G;H) GxGy = GGxy = Gxy and xy ~ N(G;H), 10 while Gx has inverse Gx -1 • Associativity in U follows from associativity in H • In fact U = N(G;H)/G. It remains to prove that (ii) of Defin­ ition 1.1 holds. Consider the set GxGy • We wish to show that GxGy = Gz for some z ~ H if and only if x ~ N{G;H) • If x é N{G;H) then GxGy = Gxy • On the other hand Gxy Ç GxGy so if GxGy = Gz for some z then Gxy = GxGy • Renee Gx 2 lxG = xG and so x ~ N{G;H) • # We shall denote the mixed group M defined in Theorem 1.3 as H/G • The nucleus of the mixed group is N{ G;H) / G and we note that for G <J H H/G ie the ordinary quotient group.