UNIT 1 Groups STRUCTURE PAGE NO. 1.1 Introduction 01 1.2 Objective 01 1.3 Normal and Subnormal Series 2-5 1.4 Composition Series 5-7 1.5 Jordan–Hölder theorem 7-11 1.6 Solvable Groups 11-13 1.7 Nilpotent Groups 13-15 1.8 Unit Summary/ Things to Remember 16 1.9 Assignments/ Activities 17 1.10 Check Your Progress 17 1.11 Points for Discussion/ Clarification 18 1.12 References 18 1.1 Introduction The theory of groups was unified starting around 1880. Since then, the impact of group theory has been ever growing, giving rise to the birth of abstract algebra in the early 20th century, representation theory, and many more influential spin-off domains. The classification of finite simple groups is a vast body of work from the mid 20th century, classifying all the finite simple groups. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic building blocks of the natural numbers. The classification of the finite simple groups is believed to classify all finite simple groups. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic building blocks of the natural numbers. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. Jordan defines isomorphism of permutation groups and proves the theorem for permutation groups. In a huge collaborative effort, the classification of finite simple groups was accomplished in 1983 by Daniel Gorenstein. 1.2 Objective After the completion of this unit one should able to : define normal, subnormal and composition series. prove Jordan–Hölder theorem. to prove fundamental theorem of Arithmetic as an application of Jordan–Hölder theorem. relate solvable and nilpotent groups. to show the solvability of finite p-group. to analyze abstract and physical systems in which symmetry is present. 1.3 Normal and Subnormal series 1 An important part in group theory is played by subnormal, normal and central series. These series are related with the view of a series of a group G, which gives insight into the structure of G. The results hold for both abelian and non abelian groups. Subgroup series: A subgroup series is a finite sequence of subgroups of a group G contained in each other such that: {e}= G0 G1 G2 G3 ...........Gn G or {e}=G0 G1 G2 G3 ........... Gn G ……………(1) One also considers infinite chains of imbedded subgroups (increasing and decreasing), which may be indexed by a sequence of numbers or even by elements of an ordered set. Here n is called as the length of the series. A subgroup series (1) is called subnormal series or sub invariant series. If also each subgroup Gi , i = 0,1,2,……n, is normal in G, the subgroup series (1) is called a normal series or invariant series in G. It is noted that for abelian groups the notions of subnormal and normal series coincide, since every subgroup is normal. So a normal series is a subnormal series, but the converse is not true in general. Sometimes the phrase "normal series" is used when the distinction between a normal series and a subnormal series is not important. Most of the authors use the term normal series, instead of subnormal series. Examples: 1. {0} < 8 < 4 < : Normal series 2. {0} < 9 < 3 < : Normal series 3. : Subnormal series. Since is not normal in . Here is the group of symmetries of the square denotes rotations and denotes mirror images. 2 Refinement of a normal series A subnormal ( normal) series {1} = K0 = K1 = · · · = Km = G is a refinement of subnormal ( normal) series {1} = H0 = H1 = · · · = Hn = G if { Hi } { Kj } . A refinement of a normal series is another normal series obtained by adding extra terms. Examples: 1. Sn > An > 1 is a normal series for G = Sn for any n ≥ 1. The associated subquotients are Sn /An ∼ /2 and An /1 = An . For n = 4 this normal series has a refinement: S4 > A4 > K > 1. 2. The series { 0 } < 72 < 24 < 8 < 4 < is a refinement of a series { 0 } < 72 < 8 < Schreier Refinement Theorem: Any two subnormal series of a group have equivalent refinements. Proof Consider the subnormal series of a group G. G G0 G1 G2 ... Gs (e) … (1) G H0 H1 H 2 ... H1 (e) … (2) Since for any i,j,k and l, Gi1 is a normal subgroup of Gi and H k1 is a normal subgroup of H k ' we get Gi, j Gi1 (Gi H j ),(i 0,1,2,...,s 1; j 0,1,2,...,t) …(3) H k,l H k1 (H k G1 ),(k 0,1,2,...,t 1;l 0,1,2,...s) …(4) are subgroup of G. 3 Now H jl is normal subgroup of H j implies that Gi' j1 is normal subgroup ofGi' j' . Similarly H k 'l1 is normal subgroup of H k,l . Since H t (e)and H 0 G, we have Gi,t Gi1 (Gi Ht ) Gil (e) Gi1 . Also Gi,0 Gi1 (Gi H0 ) Gi1 (Gi G) Gi1Gi Gi . Thus Gi,t Gi1 Gi1,0i 0,1,2,...s 1. …(5) Similarly H k,s H k1 H k1,0i 0,1,2,...,t 1 …(6) Consider two series G G0 G0,0 G0,2 ... G0,t ( G1 G1,0 ) G1,1 G1,2 …(7) ... G1,t ( G2 G2,0 ) ... Gs 1,0 ...Gs1,t Gs (e) G H0 H0,0 H0,1 H0,2 ... H0,s ( H1 H1,0 ) H1,1 H1,2 ... H1,s ( H 2 H 2,0 ) ... Ht1,0 Ht1,0 Ht1,1 ... Ht1,s Ht (e) ......(8) Both (7) and (8) have same number of terms. Clearly G0 occurs in (7) and for each m 1,2,...s,as Gm Gm1,t by (5), we see that each Gm occurs in (7) for all m. Thus (7) is a refinement of (1). Similarly (8) is a refinement of (2). Now by Zassenhaus theorem (Let A and B be any two subgroups of a group G, A* and B* be B *(B C) C *(C B) normal subgroups of A and B respectively then ) B *(B C*) C *(C B*) G G (G H ) H (H G ) H We have r,s r1 r s s1 s r s,r Gr,s1 Gr1(Gr H s1 H s1(H s Gr1) H s,r1 For all r 0,1,2..., s 1 and for all s 0,1,2,...t 1. 4 Thus (7) and (8) are equivalent. This proves the theorem. Lemma If G is a commutative group having a composition series then G is finite. Proof We firstly show that a simple abelian group must be a cyclic group of prime order. This follows the fact that any subgroup of an abelian group is normal. Now let G G0 G1 G2 ... Gs (e) be a composition series of G. Gs 1 Gs1 Since Gs1 is simple abelian, o(Gs2 ) ps1 ' where ps1 is a prime number. Gs (e) Gs2 Gs2 Further is simple abelian o ps2 , for some prime number ps2 . Gs1 Gs1 Gs2 Thus o(Gs2 ) o o(Gs1 ) ps2 ps1. Gs1 Gi Proceeding in this manner we get that G has p0 p1 p2 ...ps1 elements where pi o for Gi1 i 0,1,2,...,s 1 1.4 Composition series If a group G has a normal subgroup N which is neither the trivial subgroup nor G itself, then the factor group G/N may be formed, and some aspects of the study of the structure of G may be broken down by studying the "smaller" groups G/N and N. If G has no such normal subgroup, then G is a simple group. A composition series of a group G is a subnormal series {1} = H0 = H1= H2 = · · · = Hn= G 5 with strict inclusions, such that each Hi is a maximal normal subgroup of Hi+1. Equivalently, a composition series is a subnormal series such that each factor group Hi+1 / Hi is simple. The factor groups are called composition factors. A subnormal series is a composition series if and only if it is of maximal length. That is, there are no additional subgroups which can be "inserted" into a composition series. Thus, A subnormal series {1} = H0 = H1 = H2 = · · · = Hn = G is a composition series if and only if (1) H0 < H1 < H2 < · · · < Hn = G. (2) For each i = 0, . , n - 1, we have that there is no normal subgroup of Hi+1 that lies strictly between Hi and Hi+1. The length n (the number of subgroups in the chain, not including the identity) of the series is called the composition length. If a composition series exists for a group G, then any subnormal series of G can be refined to a composition series, informally, by inserting subgroups into the series up to maximality. Every finite group has a composition series, but not every infinite group has one. For example, the infinite cyclic group has no composition series. Example: Let G = 6 = {[0], [1], [2], [3], [4], [5]} be the cyclic subgroup of order 6 and consider the subgroups H = [2] = {[2], [4], [0]} and K = [3] = {[3], [0]}. This gives rise to the two subnormal series {[0]} = H = G, {[0]} = K = G. In fact these are both composition series. The first has composition factors H/{[0]} 3 and G/H 2, whereas the latter series has composition factors K/{[0]} 2 and G/K 3.