GENERAL I ARTICLE Beginnings of

R Sridharan

Cohomology theory is a powerful mathematical tool. This theory applied to topology is a part of , which associates algebraic invariants to topological spaces. In this article, we give a brief outline of group cohomology.

1. Introduction R Sridharan has been an Cohomology theory is a powerful mathematical tool. Adjunct Professor at the This theory applied to topology is a part of algebraic CMI, Chennai since topology, which associates algebraic invariants to topo­ retiring from TIFR, logical spaces. It can be applied equally fruitfully to Bombay in 2000 where he algebra to study the structure of algebraic objects like was a senior professor. He is also an INSA senior groups, associative algebras, etc. and this was initiated scientist now. His by Eilenberg-Mac Lane, Eckmann, Hochschild and oth­ scholarship permeates to ers. (It turns out that there is a unifying link between literature (English and these approaches, which was formalised for the first time Sanskrit) and philosophy as well. Many of the by H Cartan and S Eilenberg in their classic book H 0- algebraists in the country mological Algebra in 1956). Eilenberg and Mac Lane were his students. published in 1946, in the Annals of , their first paper on the cohomology theory of groups. Al­ most at the same time, Eckmann also introduced the same theory. These authors were led to the cohomology Dedicated to my teacher theory of groups, with motivation from algebraic topol­ 5 Eilenberg ogy, an area in which they were specialists. Indeed, the first definition of cohomology of groups in the paper of Eilenberg and Mac Lane is based on the so-called 'ho­ mogeneous cochains' and this approach is really mod­ elled on the usual definitions in algebraic topology. On the other hand, there is another, equivalent approach to the cohomology theory of groups via the so-called 'non Keywords , factor homogeneous cochains', which was given by Eilenberg­ sets, group extensions, Hil­ Mac Lane and Eckmann and is motivated by the work bert's theorem 90, cocycles, of some earlier authors on . crossed homomorphisms.

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The first definition of In this article, we give a brief outline of the work of cohomology of mathematicians like D Hilbert, E Noether, 0 Schreier, groups in the paper R Baer (and others) in group theory and note that their of Eilenberg and work already contained motivating material for defin­ Mac Lane is based ing and studying the cohomology theory of groups in general. For instance, the 90 for the on the so-called Hilbert Theorem class of 'cyclic Galois extensions' and its generalization 'homogeneous by for 'all finite Galois extensions' is re­ cochains' and this ally to show that a certain first cohomology group, of approach is really the '' of the extension is trivial. The work modelled on the of Schreier on group extensions with abelian kernels al­ usual definitions in ready contained the definition of the second cohomology algebraic topology. of groups. The work of R Baer on the construction of obstructions to the existence of group extensions with 'non abelian kernels' contained implicitly the definition of the third cohomology. We shall discuss some of these results in as elementary a manner as is possible, assum­ ing only rudimentary knowledge of group theory. 2. Group Actions and the First Cohomology The notion of a group has played a fundamental role in mathematics. For instance, Felix Klein in his famous "Erlangen Programme", emphasized the role of groups in geometry by remarking that geometry can be thought of as the study of properties of spaces invariant under the action of certain groups. A very fruitful notion in this direction is that of a group G acting on another group A. We start with a definition: We say that a group G acts on or operates on a group A, if there is a homomorphism of G in to the group Aut(A) of all of A. (We recall that an of a group A is a bijective map of A onto A which preserves the group operation of A. The set of all automorphisms of a group A is again a group under the usual composition of maps as the group operation).

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Writing the definition of G action on A in long hand, Felix Klein in his this means that for each 9 E G, and a E A, there exists famous "Erlangen an element ga E A such that the following conditions Programme", are satisfied: emphasized the role For gl,g2 E G, and a E A, we have (glg2)a = gl(g2a). of groups in la = a, and g( a.b) = ga.gb for a, b E A. geometry by remarking that If, in particular, A is an , we shall denote geometry can be the group operation in A by '+' instead of '.', the iden­ thought of as the tity element by 0 and the inverse of a by -a. study of properties of If a group G acts on an abelian group A, we call A a spaces invariant G-. We shall now assume that A is a G-module. under the action of It is natural to ask for elements of A which are fixed certain groups. A very points for this action. We recall that an element a of A fruitful notion in this is a fixed point if G fixes it or what is the same, g.a = a direction is that of a for all 9 E G. Now, 0 is clearly a fixed point. If a and b group G acting on are fixed points, then it is immediate from our definition another group A. that a-b is also fixed, since g.(a-b) = g.a-g.b = a-b. Hence the set of fixed points of A is a subgroup of A.

As we have seen, 0 E A is a fixed point and we would like to know whether there are 'nontrivial fixed points'. We shall formulate the question in a slightly different way as follows: Let B be an abelian group on which a group G acts and let A be a subgroup of B which is stable under this action of G; that is for all a E A, and for all 9 E G, we have ga E A. It can then be seen immediately that G acts also on A which we call the action 'induced' from that of G on B. The group G then acts also on the B / A, if we set for any 9 E G and b E B/A, gb = gb, for any lift b of b E B. Suppose that there is a non zero fixed point of B / A for this action of G. We wish to find out the obstruction to 'lifting' this fixed point of B / A to a fixed point for the action of G on B. To do this, let us take a fixed

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point b E B / A and choose an arbitrary element b in B whose coset modulo A is b. Consider, for any 9 E G the element gb - b. We note that the coset of gb - b modulo A is the coset 0, since gb - b = gb - b = 0, so that gb - b E" A. Obviously, these elements are zero for all 9 E G if and only if b E B is a fixed point of G. Thus, the map G~A

Ib:g~gb-b measures the obstruction to lifting b to a fixed point of G. We also note that the map Ib : G ~ A satisfies the equation Ib(g.g') = g.lb(g') + Ib(g) for g, g' E G.

For any abelian group A on which G acts, a map I : G ~ A which satisfies I(g.g') = gl(g') + I(g), for g,g' E G is called, classically,;a crossed homomorphism or 1 - cocycle of G with values in A. Thus, given a fixed point for G in B/A, we get a 1-cocycle I : G ~ A. A 1- co cycle I : G ~ A is called a 1-coboundary if there exists an a E A such that I(g) = g.a - a, Vg E G. We note that any 1-coboundary is a 1-cocycle, since for , , ( , g,g E G,g.g a - a =g 9 a - a) + ga - a.

It is easily seen that if I, I' : G ~ A are 1-cocycles then I + I' defined by (I + I')(g) = I(g) + I' (g) is again a 1-cocycle. The 1-cocycles from G ~ A form an abelian group (denoted by Zl ( G, A)) under this binary opera­ tion. The map 0 : G ~ A, defined by O(g) = 0 is the identity element and for any I E Zl ( G, A), - I defined by (-I) (g) = - I (g) is the inverse of I. It is immedi­ ately verified that the subset Bl(G, A) of 1-coboundaries of Zl ( G, A) is, in fact, a subgroup of Zl ( G, A). The quo­ tient group Hl(G, A) = Zl(G, A)/Bl(G, A) is called the first cohomology group of G with coefficients in A. With this notation, starting with a fixed point b E B / A for g, we obtain a 1-cocycle Ib : G ~ A (and it can be seen quite easily that the class of this cocycle modulo coboundaries is uniquely determined by b). This co cycle

-o------VV\AfVv------­4 RESONANCE I September 2005 GENERAL I ARTICLE is a coboundary if and only if (by definition) 3a E A, Cohomology = = S.t. g.b - b fb(g) g.a - a so that g.(b - a) = theory can be g.b - g.a b - a, i.e., if and only if b - a is a fixed = roughly defined as point of B for the action of G. Since b - a = the b, a theory of obstruction to lifting a fixed point of B / A" for the action obstructions! of G to a fixed point of B lies precisely in the class of fb E HI ( G, A) being nontrivial. This is, incidentally, the typical role cohomology plays in general, namely to pinpoint obstructions to things from happening! Thus cohomology theory can be roughly defined as a theory of obstructions! We want to include an example, where the first coho­ mology vanishes. Let G be a and K a , whose characteristic does not divide the order of G, so that the order of G is a non zero element of K. Let V be a finite dimensional vector space over K on which G operates, in fact, as K-linear transformations. Let W be a K-subspace of V which is stable under. the action of G. Then a classical result due to Maschke asserts that there exists a supplement W' of W(Le. a subspace W' of V which is such that V = W E9 W'), which is stable under the action of G. This result can be interpreted in terms of what we said earlier as follows: For vector spaces V, V' over K, let H amK(V, V') denote the vector space of all K -linear mappings from V to V' Let G act on both V and V' For f E HamK(V, V') and x E G, let x * f be the K-linear map,

V~V' defined by v ~ x * f(v) = xf(x-Iv). Then we get an action of G on HamK(V, V'). We note that the fixed points for this action are precisely the K -linear maps from V to V' which preserve the G­ action. In particular, with the notation as above, B = HomK(V, W) is a G-module, A = HomK(V/W, W) is a

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G-submodule and the corresponding quotient is H omK (W, W) on which G acts. The identity map W --+ W is obviously a fixed point for the action of G on HomK (W, W). Then the theo­ rem of Maschke can be interpreted to mean that this can be lifted to a fixed point of H omK (V, W). If p is such a fixed point it is immediate that p : V --+ W is a linear map which commutes with the action of G, and W' = Kerp = {x E plp(x) = O} is a G-supplement of W To find such a p, let p' E H omK(V, W) be any lift of the identity map W --+ W Such a lift ex­ ists since the K-subspace W has a supplement in V Then p : V --+ W defined by v H Ib 2:xEG xp' (X-IV) commutes with the action of G on V and W For y E G and v E V, (py)(v) = Ibl 2:xEG xp' (X-IyV) = Ibl 2:zEG yzp' (z-Iv) = (yp) ( v). Also p is a lift of the identity map of W If w E W p' p(w) 2:xEG xp' (x-Iw) w (w) = wand = ,bl = since p' (x-Iw) = x-Iw, x-Iw being in W, since W is G­ stable.

3. A Precursor to Hilbert Theorem 90 We shall also give a classical theorem due to Hilbert, where the first cohomology group of a certain group van­ ishes. We shall start with a very simple special case of the theorem of Hilbert. Let us denote by C, the set of complex numbers which, as we know, is a field. It has a special automorphism, namely, conjugation (which we shall denote by a or bar), which sends any complex num­ ber z to its conjugate z, which is of order 2.( We note that for any z E C, Z = z). One has then the situa­ tion in which the group G with two elements (I, a) acts on the field of complex numbers, where I (which is the identity automorphism), fixes all complex numbers and a acts as conjugation on any complex number. Obvi­ ously, the set of fixed points of C for this action of G is

-42------~~------RE-S-O-N-AN-C-E--1-se-Pt-em--be-r--2o-o-S GENERAL I ARTICLE the field ~ of real numbers. (In fact, for Z E C, Z = z, if Galois extensions and only if Z E ~). We shall now compute as examples, occur precisely in the the cohomology groups Hl(G, C) and Hl(G, C*), where following context: C is regarded as an abelian group under addition and Take a field K of C* = C - (0) is the group of non zero complex numbers characteristic 0, take under multiplication. a polynomial with Let f : G -+ C be a 1-cocycle. Since f(1) = f(12) = coefficients in K, and l.f(1) + f(1) = 2f(1), we have f(1) = O. Thus f is define L as the uniquely determined by f(a) = Zoo We have 0 = f(1) = 'smallest field' 2 f(a ) = a·f(a) + f(a) = Zo + zo, so that Zo = -Zo; that containing Kand all is Zo is purely imaginary, so that Zo = zo/2 - (zo)/2 = the roots of this a(zo/2) - (zo/2), which shows that f is a 1-coboundary; polynomial. The that is Hl(G, C) = O. Galois group is then the group which On the other hand, let f : G -+ C* be a 1-cocycle. permutes the roots. Then by an argument as above, we have f(1) = 1 and f is determined by f(a) = woo We have 1 = f(1) = 2 f(a ) = a.f(a).f(a) = wo.wo. That is wo is a complex number of absolute value 1, that is, it is on the unit i circle. Hence WO = e (} for some real (). Hence WO = ei(}/2/ e-i(}/2 = vO/vo, with VO = e-i(}/2, which shows that f is a 1-coboundary and Hl(G, C*) = O.

The pair of fields C ~ ~ is a typical example of the general situation L ~ K , where Land K are fields for which there exists a finite subgroup G of automorphisms of L whose set of fixed elements is precisely K. Such a pair, denoted by L IK, is called a Galois extension and G the Galois group of the extension LIK. In fact, Galois extensions occur precisely in the following context: Take a field K of characteristic 0, take a polynomial with coefficients in K, and define L as the 'smallest field' containing K and all the roots of this polynomial. The Galois group is then the group which permutes the roots. The field C is the smallest field containing ~ and the roots of x 2 + 1. Then I fixes the roots i and -i, whereas a(i) = I = -i !

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The study of Hilbert was entrusted by the German Mathematical So­ cohomology of the ciety in 1893, with the task of preparing an up-to-date Galois group of report on the then state of 'Algebraic ' Hilbert undertook this task and came out in 1897 with Galois extensions his magnum opus Zahlbericht, which has since became is one of the most a classic of all times. In his book, Hilbert proves as an fruitful areas of auxiliary result his Theorem 90, which shows that the research till today! first cohomology group HI ( G, L *) vanishes for the class of Galois extensions L / K for which the Galois group G is cyclic. It is not difficult to give a proof of this result lnuch like the proof we gave for the special case CjR. Later, Emmy N oether showed that this result holds more generally for any Galois extension. The study of cohomology of the Galois group of Galois ex­ tensions is one of the most fruitful areas of research till today! 4. Group Extensions, Factor Sets and the Second Cohomology Having indicated the genesis of the (first) cohomology group of a group, let us now turn to the definition of the second cohomology group which also goes back in time. The second cohomology group of G is defined as the quo­ tient group of the group of 2-cocycles by the subgroup of 2-coboundaries. The 2-cocycles have been studied under the name of 'factor sets' by 0 Schreier in the nineteenth century. These arose in the theory of 'group extensions'. We shall briefly explain first the problem of group ex­ tensions. We shall ask ourselves the (somewhat vague) question: "How far does the knowledge of a normal sub­ group K of a group E and the corresponding quotient E / K determine the group E 7" Let us explain the sit­ uation a little precisely. Let K be a normal subgruop of a group E and let us write Q = E / K. For each coset x E Q, let us pick a coset representative x E E. This gives an assignment Q --+ E, which we shall denote by t. We shall take for the coset containing the identity, its most natural coset representative namely the iden-

44------~------R-ES-O-N-A-N-C-E-I-s-e-Pt-e-m-be-r--2-oo-S GENERAL I ARTICLE tity element 1 of E, that is tel) = 1. If we denote by 2-cocycles have been TJ : E 4 E / K the canonical homomorphism which maps studied under the E E any element x E to its coset x Q, we obviously have name of 'factor sets' TJ 0 t t : 4 = la. Such a map Q E is c~lled a set the­ by 0 Schreier in the oretic 'section' of TJ. It is not true in general that t is a nineteenth century. homomorphism of groups and the obstruction to t being These arose in the a homomorphism, in fact, gives a 2-cocycle as we shall theory of 'group see presently. extensions' . Let us recall that for any group K, there are the spe­ cial automorphisms of K, namely the inner automor­ phisms of K , that is those automorphisms of the form k ~ kokko I, for some ko E K (this inner automor­ phism is denoted by lnt(ko)). The inner automorphisms form a normal subgroup denoted by I nt K of the group AutK of all automorphisms of K. In fact, for ko, lo E K,lnt(ko) 0 Int(lo) = Int(ko.lo), and Int(ko)-I = Int (kOI) and if Int(ko) is an and a is an arbirary automorphism, we have ao Int(ko) 0 a-I = I nt (a ( ko)) . Let us define for x E Q, and k E K, x.k = t(x).k.t(X)-l. We thus get for each x E Q, an automorphism of K. We claim that this assignment gives rise to a homomorphism cP : Q 4 AutK/ I ntK independent of t. Indeed if t(x) and t'(x) are two coset representatives of x E E, we have t'(x) = kot(x) for some ko E K, so that t'(X}.k.t'(X)-1 = ko.t(x).k.t(X)-I.(ko)-1 =

Intko(t(x).k.t(x)-I) and so that the map cP : Q -+ AutK/ I ntK is indepen­ dent of t. It is easily checked that cP is a homomorphism.

The map cP can also be described in the following way: For e E E the map k ~ e.k.e-I of K onto itself is an automorphism, so that we have a map E 4 AutK, which is easily seen to be a homomorphism. Under this

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8aer's study of homomorphism any element l E K maps into the in­ extendability of ner automorphism of I nt(l) of K. We therefore have kernels found its a homomorphism Q = ElK --+ AutKllntK which is systematic precisely the map if;. treatment in the Suppose Q is a group and K is a group with a homo­ work of Eilenberg­ morphism if; : Q --+ AutKI I ntK. Following Baer and Mac Lane. Eilenberg-Mac Lane we call the pair (K, ~) a kernel. As we just now noticed, once there exists a group E containing K as a normal subgroup with Q as the cor­ responding quotient ElK, then we get a kernel (K, if; ) which we shall call an extendable kernel. One could then ask whether any kernel arises in this way that is whether any kernel is extendable. Baer in 1934 stud­ ied this question and showed by a counterexample that not all kernels are extendable. The obstruction to the extendability of a kernel lies in a certain third cohomol­ ogy class of Q with values in th~ center of K being non trivial. This study which began with Baer found its sys­ tematic treatment in the work of Eilenberg-Mac Lane. We shall not go into this topic in detail.

'. - Suppose now that there exists a homomorphism if; : Q --+ AutK, such that the composite homomorphism Q --+ AutK --+ AutKLlntK is if; i.e. if; can be 'lifted' to a homomorphism if; : Q --+ AutK. We then get an action of Q on K by setting for ,,\ E Q and k E K, )".k = if;()")(k). The question that one would like to ask and answer is the following question: Given a group K, and a group Q which acts on K, is there a group E which contains K as a normal subgroup and with Q isomorphic to ElK i.e. whether (K, if;) is extendable. We shall start with a 'trivial' solution to the question above. If K and Q are given as above, we take E = Q ~ K, the so-called ' , of Q and K, defined as follows: We' take the set .E = Q x K: the cartesian pr~duct of Q and K and define the binary operation on E by setting: for (x,a),(y,b) E E,

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(x, a).(y, b) = (xy, a.(x.b)). It is easily verified that E is a group with identity (1,0) and for (x, a) E E, (x, a)-l = (x-I, -x-l.a). It is also easy to check that K can be identified with the subgroup {(I, a)la E K} of E which is, in fact, a normal subgroup of E and the map '1] : (A, a) t-t A induces an isomorphism of E/ K onto Q. (Note that if Q operates trivially on K, i.e., x.b = b for x E Q and b E K, then the group operation on E gives simply the direct product of the groups Q and K). (We note that Q can also be identified as a subgroup {(A,O)IA E Q} of E and that Q is not, in general, a normal subgroup of E). The map t : A t-t (A,O) of Q --t E is a homomorphism with '1] 0 t = IQ , that is, t is a section. Conversely if E is a group with K as a normal subgroup, '1] : E --t E/ K ::;: Q, the canonical homomorphism such that there is a section t : E / K --t E which is also a homomorphism Q to E, then the map E --t Q ~ K given by e t-t ('1] (e) , t('1](e))-l.e) is easily checked to be an isomorphism of the group E with Q ~ K and whose inverse is given by (A, a) t-t t(A).a. What we have said earlier holds in particular if K = A is an abelian group (in which case we shall denote the group operation in A by +). We shall assume this from now on. We have, therefore, produced a solution of the question we started with. What about all solutions? Let E be a group, A a normal abelian subgroup, Q = E / A and '1] : E --t Q the natural homomorphism. Let now t : Q --t E be a set theoretic section with t(!) = 1. Note that t is not necessarily a homomorphism and if it is, as we have seen above, E is isomorphic to the semidirect product Q ~ A. _Let us now write down the obstruction to the section t from being a homomorphism. Let for A, J-l E Q, t(A).t(J-l) = I(A, J-l).t(A.J-l) for some I(A, J-l) E A which follows from the fact '1]( t( A) .t(J-l) .t( A.J-l )-1) = '1]( t( A)) .17( t(J-l)).",( t( AJ-l) -1 ) = A.J-l.(AJ-l)-l = 1.

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Thus, I(A, p,) = t(A)t(p,)t(Ap,)-1 E Ker'fJ = A. Now the condition that the multiplication in Q is associative gives, for A, p" v E Q t( A)t(p, )t( v) = (t(A)t(p,))t(v) (/(A, p,)t(A.p,))t(v) - I(A, p,)(t(Ap,)t(v)) I(A, p,)/(Ap" V)t(Ap,V) - t( A)( t(p, )t(v)) t(A)(/(p" v).t(p,v)) - t(A)/(p" V).t(A)-1.t(A).t(p,V) - A./(p" v)/(A, p,V)t(Ap,V).

Equating the expressions on the right hand sides and cancelling t(AJ-tV), we get, for A, p" v E Q

A·/(p" v) - I(Ap" v) + I(A, p,v) - I(A, p,) = o. (*) Let Q be a group and A a Q-module. A map I : Q x Q ~ A is called a 2-cocycle ( or a lactor set as it was classically called), if it satisfies the condition (*) above. We say that a 2-cocycle I is normalized if I(A, p,) = 0 for A = 1 or p, = L Using (*), it is easily verified that I is normalized if and only if 1(1, 1) = O. Thus, given a group E and an abelian normal subgroup A of E, with Q = E / A, we get a normalized 2-cocycle I : Q x Q ~ A whose definition depends, of course, upon the choice of a set theoretic section t : Q ---+ E. Let t' : Q ~ E be another section. Then for any A E Q, t'(A) = h(A)t(A) for some h{ A) E A, for all A E Q so that we have a map h : Q ~ A. We note that h{I) = 1 since t' (1) = t(I) = 1. Any map h : Q ~ A is called a I-cochain. Then if I' is the 2-cocyc1e defined through the choice of the section t', we have,

I'(A, p,) = t'(A)t'(p,)t'{Ap,)-1 _ h(A)t{A)h{p,)t{p,)t( Ap, )-1 h{ Ap, )-1 - h(A) + t(A)h(p,)t{A)-1 + t{A)t{p,)t(Ap,)-1 + h(Ap,)­ h{A) + Ah(p,) - heAp,) + I(A, p,).

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For a 1-cochain h : Q ~ A, the map 6h : Q x Q ~ A defined by 6h(>", J-L) = >..h(J-L) - h(>"J-L) + h(>") is veri­ fied to satisfy the condition (*) above, so that it is a 2-cocycle. It is a special kind of 2-cocycle, called a 2- coboundary. The set of 2-cocycles Q x Q ~ A becomes an abelian group, denoted by Z2(Q, A), if we define, for f,g E Z2(G, A), (f + g) by

(f + g)(>.., J.L) = f(>", J.L) + g(>.., J.L).

The set of 2-coboundaries is a subgroup of Z2 (Q, A) de­ noted by B2(Q, A). The quotient Z2(Q,A)/B2(Q,A) is called the second cohomology group of Q with values in A, and denoted by H2(Q, A). Given any 2-cocycle f, there exists a normalized 2-cocycle j such that the class of [f] of f in H2 (Q, A) = the class [}] of f; in fact if h : Q ~ A is defined by h(>") = >..f(l, l) then j = f - 6(h) is normalized. To summarise what we have done so far: if A is an abelian normal subgroup of a group E and Q = E/A, then A is a Q-module and there exists through the choice of a section t : Q ~ E, an element f E Z2 (Q, A) which is a normalised 2-cocycle whose class in H2 (Q, A) is independent of t. The remarkable fact is that conversely given a group Q, a Q-module A and a class [f] E H2(Q, A), there exists a group E containing A as a normal subgroup and Q = E / A such that the class [f] occurs as above. This is achieved as follows:

Let f E Z2 (Q, A) be a representative of [f] which we can, without loss of generality, assume to be normalised. We take E(f) = Q x A, the cartesian product of Q and A and define a binary operation on E(f) as follows: For (>.., a) and (J.L, b) in E, we define

(>.., a).(J.L, b) = (>"j.L, a + >"b + f(>", J.L)).

(Note that if f = 0 this definition gives the semidirect product defined above). It' is easy to check that (1, 0) is

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The theorem which the identity element; in fact (A, a).(I, 0) = (A, a + A.O + solves the problem of I(A,I)) = (A, a) since I(A, 1) = 0, 1 being normalized. extensions where the Similarly we have (1, O)(A, a) = (A, a) since 1(1, A) = o. Also we see that every element has an inverse. The normal subgroup A is crucial condition to check is that this binary operation abelian, was proved on E(/) is associative. We have, by Schreier and has been used (A, a)«JL, b)(v, c)) (A, a)(JLv, b + JLC + I(J-L, v)) extensively by (AJ-LV, a + Ab + AJLC + A/(JL, v)+ Brauer, Noether and I(A, J-Lv). Albert in the study of finite dimensional division algebras. «A, a)(JL, b))(v, c) (AJ-L, a + Ab + I(A, J-L))(v, c) (AJLV, a + Ab + I(A, J-L) + AJLC+ I(AJL, v). Looking at the right hand sides we see that the asso­ ciativity of the operation is equivalent to the condition that I is a 2-cocycle. Suppose I and I' are two normalized 2-cocycles both representing [I], then E(/) and E(/') as constructed above are isomorphic. In fact if I' = I + 8h with h : Q -4 A, a I-cochain which we can assume to satisfy h(I) = 0, then the map E(I') -4 E(/) given by (A, a) 1-+ (A, a + h(A)) is an isomorphism. Let Q be a group, A a Q-module, and E a group, which has A as a normal abelian subgroup with quotient iso­ morphic to Q. If I is a normalized 2-cocyc1e of Q with values in A, given by the choice of a section Q --+ E, then it is easy to check that E ~ E(/). Putting all these together we get the following: Theorem: Let Q be a group; if A is a Q-module, then the group H2(Q, A) parametrises 'upto isomorphism' all groups E which have A as a normal subgroup and Q as the corresponding quotient. This theorem which solves the problem of extensions where the normal subgroup A is abelian, was proved

-50------~~------R-E-SO-N-A-N-C-E-I-s-ep-te-m-b-er-.-2-00-5 GENERAL I ARTICLE by Schreier and has been used extensively by Brauer, A theorm due to Noether and Albert in the study of finite dimensional Schur and division algebras. Zassenhaus says We end this section with a theorm due'to Schur and that the problem of Zassenhaus which says that the problem of extensions extensions is 'trivial' is 'trivial' in case the normal subgroup and the corre­ in case the normal sponding quotient are finite groups of coprime order. subgroup and the We have, more precisely, the following: corresponding quotient are finite Theorem: Let E be a group which has a normal sub­ groups of coprime group K of order m and let the quotient group E / K order. have order n with m and n coprime. Then E is isomor­ phic to the semidirect product E / K I>< K. The proof of the above theorem is achieved by reducing to the case where A is abelian by an inductive argument. 5. Eilenberg-Mac Lane and Eckmann's Coho­ mology of Groups The considerations of the previous sections show that the notion of the first, second (and to some extent the third) cohomology groups already existed classically. What Eilenberg-Mac Lane, and independently Eckmann did was to give the definition of the cohomology groups Hn(Q, A) for all n for a group Q and any Q-module A. Let us very briefly recall their definition. For any integern ~ 0, let Qn denote ,the cartesian product of a group Q, n- times.. An n-cochain is by definition any map f : Qn -+ A. (A O-cochain is by convention an element of A). We define the ~oboundary of an n­ cochain as the (n+1)-cochain 8(/) : Qn+1 --t A defined by <5(f)(91, .... , 9n+l) = 91f(92, .. ~9n+l)+ f(91·92, ... , 9n+d + ... + (-1 )i+1 f(91, ... , 9i.gi+1, ... 9n+l) + ... + (_1)n+1 f(91, ... gn). An n-cochain Qn -+ A is called an n-cocycle if <5(/) = o. If / and 9 are n-cocycles then / + 9 defined by (f + g)(g1,g2, ,gn) = f(91, ,gn) + 9(g11 ,gn) is again

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an n-cocyle and the set of n-cocyles is an abelian group for this operation and is denoted by zn ( Q, A). An n­ cochain is called a n-cobaundary if there exists an (n-l)­ cochain 9 : Qn+l -t A such that f = 8(g). It is easily checked .(for example by a direct computation) that any n-coboundary is an n-cocycle. Then the set of n-coboundaries denoted by Bn(Q, A) is a subgroup of zn(Q, A) and the quotient Hn(Q, A) = zn(Q, A)/ Bn (Q, A) is called the n-th cohomology group of Q with coefficients in A. It follows from the very definition that for n=l and 2 we get backthe cohomology groups defined earlier. Cohomology theory of groups has played an important role in our understanding of group theory, has also pro­ Address for Correspondence vided an impetus for the introduction of homological R Sridharan methods in algebra and has innumerable applications. Chennai Mathematical Institute Acknowledgements 92 G.N. Chetty Road T. Nagar I am very grateful to K R Nagarajan and T Saravanan, Chennai 600017, India. who went through the article meticulously and corrected Email:[email protected] the mistakes.

"The most beautiful experience we can have is the mysterious. It is the fundamen­ tal emotion that stands at the cradle of true art and true science. Whoever does not know it and can no longer wonder, no longer marvel, is as good as dead, and his I I' ~ eyes are dimmed. It was the experience of mystery - even if mixed with fear - that engendered religion. A knowledge of the existence of something we cannot penetrate, our perceptions of the profoundest reason and the most radiant beauty, which only in their most primitive forms are accessible to our minds: it is this knowledge and this emotion that constitute true religiosity. In this sense, and only this sense, I am a deeply religious man ... I am satisfied with the mystery oflife's eternity and with a knowledge, a sense, of the marvelous structure of existence - as well as the humble attempt to understand even a tiny portion of the Reason that manifests itself in nature." Albert Einstein

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