DOCSLIB.ORG

Math 3560 HW Set 9

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math 3560 HW Set 9 Math 3560 HW Set 9 Kara November 5, 2013 4 2 3 s 3 4 (14.1) The conjugacy classes of D5 are feg; fr; r g; fr ; r g; fs; rs; r ; r s; r sg. (14.4) Conjugacy classes in S6 are formed by permutations of the same cycle structure. There are exactly 11 cycle structures in S6 and all permutations with a given structure form one conjugacy class. Here, there is a list of representatives of 11 possible cycle struc- tures: e, (12), (123), (1234), (12345), (123456), (12)(34), (123)(45), (123)(456), (1234)(56), (12)(34)(56). Using the algorithm in the book(pages 76, 77), we find g = (156423) satisfying the equation g(123)(456)g−1 = (531)(264). This g is odd. But this is not the only element satifying the equation above. Note that (531)(264) = (264)(531) and h = (126)(345) is even and satisfies h(123)(456)h−1 = (264)(531):(use the same algorithm for (264)(531)). We can easily find h 2 S8 conjugating α = (12345)(678) to β = (43786)(215), for example h = (14856237). This h is odd, but it is not enough to claim that α and β are not conjugate in A8. There might be other elements conjugating α to β, maybe not all of them are odd. In fact all of them will be odd - but this is not trivial and requires a proof. Assume there is g 2 A8 such that gαg−1 = β = hαh−1 where h = (14856237): This means h−1gα = αh−1g i.e. h−1g, which is an odd permutation being a product of an even and an odd permutations, commutes with α. Now, we will show that any permutation which commutes with α must be even. This will give us a contradiction which means that α and β cannot be conjugate in A8. Assume that γ commutes with α: γα = αγ i.e. γαγ−1 = α Note that: α(γ(6)) = γαγ−1(γ(6)) = γα(6) = γ(7); α(γ(7)) = γαγ−1(γ(7)) = γα(7) = γ(8); α(γ(8)) = γαγ−1(γ(8)) = γα(8) = γ(6): Thus (γ(6); γ(7); γ(8)) is a 3-cycle in α. We have exactly one 3-cycle in α, (678) = (786) = 1 (867). Therefore γ(6) can be 6,7 or 8. If γ(6) = 7, then (γ(6); γ(7); γ(8)) = (786) which means γ contains the 3-cycle (678). If γ(6) = 8, then (γ(6); γ(7); γ(8)) = (867) which implies that γ contains the 3-cycle (687) = (678)2. The remaining case corresponds to the case where γ contains e = (678)3. So, we see that γ contains a 3-cycle which is a power of (678). A similar argument gives that γ also contains a 5-cycle which is a power of (12345). Henece, γ is an even permutation since powers of 3-cycles and 5-cycles are even(so is their product). (14.5) To prove. The 3-cycles in A5 form a single conjugacy class. Proof. Let x; y be 3-cycles in A5. The 3-cycles in S5 form a single conjugacy class. There- −1 fore, there exists g 2 S5 such that gxg = y. If g is even, then we are done, so suppose g is odd. Without loss of generality, suppose x = (123). Then (45)x(45) = x because x and (45) are disjoint. Therefore, letting h = g(45), hxh−1 = g(45)(123)(45)g−1 = g(123)g−1 = gxg−1 = y: Furthermore, since g and (45) are odd, h is even. Hence, x is conjugate to y. (14.6) Let S be a set of 8 elements, and suppose fUig is a partition of S such that fUig has three sets, two of which have 3 elements and the third of which has 2 elements. An element, σ, of S8 preserves the partition if for all j 2 S, j 2 Ui if and only if σ(j) 2 Ui. Then there are 2 · 2 · 1 = 4 elements of S8 which preserve a given partition. The number of elements of S8 with cyclic structure (12)(345)(678) is equal to the product of the number of ways to partition a set of 8 into a set of 2 and two sets of 3 and the 8 number of elements which preserve a partition. There are 2 ways to choose the set of 2, 6 and from the remaining 6 elements, there are 3 ways to choose another set of 3. However, 1 6 since we do not distinguish between the two sets of 3, there are 2 3 ways to partition a set of 6 into two sets of 3. Therefore, there are 8 16 4 · · = 1120 2 2 3 elements of S8 with cyclic structure (12)(345)(678). (12 : : : n−1) 2 An, so s(12 : : : n−1) = (12 : : : n − 1)s, which implies s(n) = n and s(i + 1) = s(i) + 1 mod n − 1 for 1 ≤ i < n. s(1)−1 Thus, s = (12 : : : n − 1) . Noting that An−1 can be viewed as a subgroup of An, we can use the same argument to s how s = e. (14.11) Denote by Fj matrix obtained from I by multiplying j-th row by −1, and by Eij matrix obtained from I by switching i-th and j-th row. Both of this operations will change determinant of I by multiplying it by −1. Thus all these matrices have nonzero determinant, so they are invertible. 2 We are to find center of GLn(R). Call it Z. If a matrix A is in Z then in particular it commutes with all the matrices Fj, Eij. As usual denote (i; j)-th entry of A by aij. Note that AFj is matrix A with j-th column multiplied by −1, while Fj A is matrix A with j-th row multiplied by −1. Therefore AFj = Fj A gives that for all k 6= j, −akj = akj. Varying j we get that A must be a diagonal matrix. Now notice that AEij is matrix A with i and j-th columns switched, while Eij A is matrix A with i and j-th row switched. Analyzing the (i; j)-th entry in AEij and in Eij A gives us equation aii = ajj. Again, varying i; j we obtain that, to commute with all the matrices Fj, Eij, our matrix A needs to be diagonal, with same entry on diagonal, that is A = aI for some a 6= 0 (as we want A to be in GLn(R)). Such matrices commute with every element of GLn(R). Therefore we have proved that center of GLn(R) is fa I j a 2 R n f0gg. (15.3) The order of the quaternion group Q is 8 and the order of any subgroup of Q divides 8. Hence, the orders of possible non-proper subgroups of Q are 2 and 4. The only element of Q of order 2 is -1, and all the other elements except the identity have order 4. Hence, H = f1; −1g is the only subgroup of Q with 2 elements. Since both 1 and -1 commute with every element of Q, xH = Hx for every x 2 Q. Thus, H is normal by Theorem 15.3. All the other subgroups of Q with 4 elements are also normal by Theorem 15.4 since the index of such a subgroup in Q is equal to 2. (15.6) To prove. If H; K are normal subgroups of a group, and H \ K = feg, then hk = kh for all h 2 H, k 2 K. Proof. Since H and K are normal, kH = Hk and hK = Kh for all h 2 H, k 2 K. Then there exist h0 2 H, k0 2 K such that kh = h0k; hk−1 = k0h: This implies khk−1 = h0; khk−1 = kk0h; so combining these, we have h0 = kk0h, or equivalently, h0h−1 = kk0. However, h0h−1 2 H, kk0 2 K, and H \ K = feg. Therefore, h0h−1 = e, so h0 = h, which implies kh = hk, as desired. (15.12) A proper normal subgroup of A4 is fe; (12)(34); (13)(24); (14)(23); g (see page 80 in the book). The commutators are: (12345)−1(345)−1(12345)(345) = (245) 3 (12)(34)−1(345)−1(12)(34)(345) = (354) Let H be a normal subgroup of A5. It can contain only even elements so only 5-cycles, 3-cycles, or composition of disjoint transpositions. If it contains a 5-cycle, then we can relabel elements if needed and assume this 5-cycle is a = (12345). Note that a−1 also is in H, and any cojungate of a is in H, and products of these. Therefore its commutator with (345) needs to be in H. Our computations show that it is a 3-cycle. Thus if H contains 5-cycle, it needs to contain a 3-cycle as well. Similarly, if H contains composition of disjoint transpositions, we can assume that it is (12)(34) and use above computations to deduce H contains a 3-cycle - the commutator of (12)(34) and (345). In A5, 3-cycles forms one conjugacy class, therefore H needs to contain all 3-cycles. From theorem 6.5 we know that 3-cycles generate all A5. Thus H = A5. 4.
Load more

Recommended publications
  • Mathematics of the Rubik's Cube
    Mathematics of the Rubik's cube Associate Professor W. D. Joyner Spring Semester, 1996{7 2 \By and large it is uniformly true that in mathematics that there is a time lapse between a mathematical discovery and the moment it becomes useful; and that this lapse can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful." John von Neumann COLLECTED WORKS, VI, p. 489 For more mathematical quotes, see the first page of each chapter below, [M], [S] or the www page at http://math.furman.edu/~mwoodard/mquot. html 3 \There are some things which cannot be learned quickly, and time, which is all we have, must be paid heavily for their acquiring. They are the very simplest things, and because it takes a man's life to know them the little new that each man gets from life is very costly and the only heritage he has to leave." Ernest Hemingway (From A. E. Hotchner, PAPA HEMMINGWAY, Random House, NY, 1966) 4 Contents 0 Introduction 13 1 Logic and sets 15 1.1 Logic................................ 15 1.1.1 Expressing an everyday sentence symbolically..... 18 1.2 Sets................................ 19 2 Functions, matrices, relations and counting 23 2.1 Functions............................. 23 2.2 Functions on vectors....................... 28 2.2.1 History........................... 28 2.2.2 3 × 3 matrices....................... 29 2.2.3 Matrix multiplication, inverses.............. 30 2.2.4 Muliplication and inverses...............
    [Show full text]
  • Chapter 3: Transformations Groups, Orbits, and Spaces of Orbits
    Preprint typeset in JHEP style - HYPER VERSION Chapter 3: Transformations Groups, Orbits, And Spaces Of Orbits Gregory W. Moore Abstract: This chapter focuses on of group actions on spaces, group orbits, and spaces of orbits. Then we discuss mathematical symmetric objects of various kinds. May 3, 2019 -TOC- Contents 1. Introduction 2 2. Definitions and the stabilizer-orbit theorem 2 2.0.1 The stabilizer-orbit theorem 6 2.1 First examples 7 2.1.1 The Case Of 1 + 1 Dimensions 11 3. Action of a topological group on a topological space 14 3.1 Left and right group actions of G on itself 19 4. Spaces of orbits 20 4.1 Simple examples 21 4.2 Fundamental domains 22 4.3 Algebras and double cosets 28 4.4 Orbifolds 28 4.5 Examples of quotients which are not manifolds 29 4.6 When is the quotient of a manifold by an equivalence relation another man- ifold? 33 5. Isometry groups 34 6. Symmetries of regular objects 36 6.1 Symmetries of polygons in the plane 39 3 6.2 Symmetry groups of some regular solids in R 42 6.3 The symmetry group of a baseball 43 7. The symmetries of the platonic solids 44 7.1 The cube (\hexahedron") and octahedron 45 7.2 Tetrahedron 47 7.3 The icosahedron 48 7.4 No more regular polyhedra 50 7.5 Remarks on the platonic solids 50 7.5.1 Mathematics 51 7.5.2 History of Physics 51 7.5.3 Molecular physics 51 7.5.4 Condensed Matter Physics 52 7.5.5 Mathematical Physics 52 7.5.6 Biology 52 7.5.7 Human culture: Architecture, art, music and sports 53 7.6 Regular polytopes in higher dimensions 53 { 1 { 8.
    [Show full text]
  • Finiteness of $ Z $-Classes in Reductive Groups
    FINITENESS OF z-CLASSES IN REDUCTIVE GROUPS SHRIPAD M. GARGE. AND ANUPAM SINGH. Abstract. Let k be a perfect field such that for every n there are only finitely many field extensions, up to isomorphism, of k of degree n. If G is a reductive algebraic group defined over k, whose characteristic is very good for G, then we prove that G(k) has only finitely many z-classes. For each perfect field k which does not have the above finiteness property we show that there exist groups G over k such that G(k) has infinitely many z-classes. 1. Introduction Let G be a group. A G-set is a non-empty set X admitting an action of the group G. Two G-sets X and Y are called G-isomorphic if there is a bijection φ : X → Y which commutes with the G-actions on X and Y . If the G-sets X and Y are transitive, say X = Gx and Y = Gy for some x ∈ X and y ∈ Y , then X and Y are G-isomorphic if and only if the stabilizers Gx and Gy of x and y, respectively, in G are conjugate as subgroups of G. One of the most important group actions is the conjugation action of a group G on itself. It partitions the group G into its conjugacy classes which are orbits under this action. However, from the point of view of G-action, it is more natural to partition the group G into sets consisting of conjugacy classes which are G- arXiv:2001.06359v1 [math.GR] 17 Jan 2020 isomorphic to each other.
    [Show full text]
  • Math 3230 Abstract Algebra I Sec 3.7: Conjugacy Classes
    Math 3230 Abstract Algebra I Sec 3.7: Conjugacy classes Slides created by M. Macauley, Clemson (Modified by E. Gunawan, UConn) http://egunawan.github.io/algebra Abstract Algebra I Sec 3.7 Conjugacy classes Abstract Algebra I 1 / 13 Conjugation Recall that for H ≤ G, the conjugate subgroup of H by a fixed g 2 G is gHg −1 = fghg −1 j h 2 Hg : Additionally, H is normal iff gHg −1 = H for all g 2 G. We can also fix the element we are conjugating. Given x 2 G, we may ask: \which elements can be written as gxg −1 for some g 2 G?" The set of all such elements in G is called the conjugacy class of x, denoted clG (x). Formally, this is the set −1 clG (x) = fgxg j g 2 Gg : Remarks −1 In any group, clG (e) = feg, because geg = e for any g 2 G. −1 If x and g commute, then gxg = x. Thus, when computing clG (x), we only need to check gxg −1 for those g 2 G that do not commute with x. Moreover, clG (x) = fxg iff x commutes with everything in G. (Why?) Sec 3.7 Conjugacy classes Abstract Algebra I 2 / 13 Conjugacy classes Proposition 1 Conjugacy is an equivalence relation. Proof Reflexive: x = exe−1. Symmetric: x = gyg −1 ) y = g −1xg. −1 −1 −1 Transitive: x = gyg and y = hzh ) x = (gh)z(gh) . Since conjugacy is an equivalence relation, it partitions the group G into equivalence classes (conjugacy classes). Let's compute the conjugacy classes in D4.
    [Show full text]
  • A STUDY on the ALGEBRAIC STRUCTURE of SL 2(Zpz)
    A STUDY ON THE ALGEBRAIC STRUCTURE OF SL2 Z pZ ( ~ ) A Thesis Presented to The Honors Tutorial College Ohio University In Partial Fulfillment of the Requirements for Graduation from the Honors Tutorial College with the degree of Bachelor of Science in Mathematics by Evan North April 2015 Contents 1 Introduction 1 2 Background 5 2.1 Group Theory . 5 2.2 Linear Algebra . 14 2.3 Matrix Group SL2 R Over a Ring . 22 ( ) 3 Conjugacy Classes of Matrix Groups 26 3.1 Order of the Matrix Groups . 26 3.2 Conjugacy Classes of GL2 Fp ....................... 28 3.2.1 Linear Case . .( . .) . 29 3.2.2 First Quadratic Case . 29 3.2.3 Second Quadratic Case . 30 3.2.4 Third Quadratic Case . 31 3.2.5 Classes in SL2 Fp ......................... 33 3.3 Splitting of Classes of(SL)2 Fp ....................... 35 3.4 Results of SL2 Fp ..............................( ) 40 ( ) 2 4 Toward Lifting to SL2 Z p Z 41 4.1 Reduction mod p ...............................( ~ ) 42 4.2 Exploring the Kernel . 43 i 4.3 Generalizing to SL2 Z p Z ........................ 46 ( ~ ) 5 Closing Remarks 48 5.1 Future Work . 48 5.2 Conclusion . 48 1 Introduction Symmetries are one of the most widely-known examples of pure mathematics. Symmetry is when an object can be rotated, flipped, or otherwise transformed in such a way that its appearance remains the same. Basic geometric figures should create familiar examples, take for instance the triangle. Figure 1: The symmetries of a triangle: 3 reflections, 2 rotations. The red lines represent the reflection symmetries, where the trianlge is flipped over, while the arrows represent the rotational symmetry of the triangle.
    [Show full text]
  • MAT301H1S Lec5101 Burbulla
    Chapter 11: Fundamental Theorem of Abelian Groups Chapter 24: Conjugacy Classes MAT301H1S Lec5101 Burbulla Week 10 Lecture Notes Winter 2020 Week 10 Lecture Notes MAT301H1S Lec5101 Burbulla Chapter 11: Fundamental Theorem of Abelian Groups Chapter 24: Conjugacy Classes Chapter 11: Fundamental Theorem of Abelian Groups Chapter 24: Conjugacy Classes Week 10 Lecture Notes MAT301H1S Lec5101 Burbulla Chapter 11: Fundamental Theorem of Abelian Groups Chapter 24: Conjugacy Classes Properties of External Direct Products Let G1; G2;:::; Gn; G; H be finite groups. Recall (from Chapter 8): 1. The order of (g1; g2;:::; gn) 2 G1 ⊕ G2 ⊕ · · · ⊕ Gn is lcm(jg1j; jg2j;:::; jgnj): 2. Let G and H be cyclic groups. Then G ⊕ H is cyclic if and only if jGj and jHj are relatively prime. 3. The external direct product G1 ⊕ G2 ⊕ · · · ⊕ Gn of a finite number of finite cyclic groups is cyclic if and only if jGi j and jGj j are relatively prime, for each pair i 6= j: 4. Let m = n1n2 ··· nk : Then Zm ≈ Zn1 ⊕ Zn2 ⊕ · · · ⊕ Znk if and only if ni and nj are relatively prime for each pair i 6= j: Week 10 Lecture Notes MAT301H1S Lec5101 Burbulla Chapter 11: Fundamental Theorem of Abelian Groups Chapter 24: Conjugacy Classes Example 1 (from Chapter 8) Z30 ≈ Z2 ⊕ Z15 ≈ Z2 ⊕ Z3 ⊕ Z5: Consequently, Z2 ⊕ Z30 ≈ Z2 ⊕ Z2 ⊕ Z3 ⊕ Z5: Similarly, Z30 ≈ Z3 ⊕ Z10 and consequently Z2 ⊕ Z30 ≈ Z2 ⊕ Z3 ⊕ Z10 ≈ Z6 ⊕ Z10: This shows that Z2 ⊕ Z30 ≈ Z6 ⊕ Z10: But Z2 ⊕ Z30 is not isomorphic to Z60; one has an element of order 60, but the other one doesn't.
    [Show full text]
  • Conjugacy Class As a Transversal in a Finite Group
    Journal of Algebra 239, 365᎐390Ž. 2001 doi:10.1006rjabr.2000.8684, available online at http:rrwww.idealibrary.com on A Conjugacy Class as a Transversal in a Finite Group Alexander Stein Uni¨ersitat¨ Kiel, 24118 Kiel, Germany View metadata, citation and similar papers at core.ac.uk brought to you by CORE Communicated by Gernot Stroth provided by Elsevier - Publisher Connector Received September 1, 2000 The object of this paper is the following THEOREM A. Let G be a finite group and let g g G. If g G is a trans¨ersal to some H F G, then² gG : is sol¨able. Remarks.1IfŽ. g G is a transversal, the coset H contains exactly one aGg F a <<<s G < s g g . Therefore H CgGŽ .. On the other hand G : H g < a < s a aGg G : CgGGŽ . ; therefore H CgŽ . for some g g . Ž.2Ifg G is a transversal, then it is both a left and a right transversal as g eh s hg eh if g, e g G, h g H and therefore G s g GH s Hg G. As an abbreviation we define: DEFINITION. Let G be a finite group and let g g G such that G s G gCGŽ. g. Call G a CCCP-group, where CCCP stands for conjugacy class centralizer product. The idea to study these groups came from a paper by Fischerwx 10 . For details of his work and the relation to Theorem A we refer to Section 1, but we will give here a short summary: Fischer defines a so called ‘‘distributive quasigroup’’ Q and a certain finite group G s GQŽ.F Aut Ž.Q .
    [Show full text]
  • Theorem: the Order of a Conjugacy Class of Some Element Is Equal to the Index of the Centralizer of That Element
    Theorem: The order of a conjugacy class of some element is equal to the index of the centralizer of that element. In symbols we say: jCl(a)j = [G : CG(a)] Proof: Since [G : CG(a)] is the number of left cosets of CG(a), we want to define a 1-1, onto map between elements in Cl(a) and left cosets of CG(a). Let H = CG(a). Since the elements of Cl(a) are conjugates of a, we define f by: f : xax−1 7−! xH This is 1-1 since if f(x) = f(y) then xH = yH and thus xh = y. But then x−1y 2 H so that x−1y commutes with a and therefore: x−1ya = ax−1y. From that we conclude that yay−1 = xax−1 proving 1-1. As for onto, just note that for any x 2 G, xax−1 is in Cl(a). However we also need to show the map is \well-defined”. I.e. we need to show that if xax−1 and yay−1 are two descriptions of the same element in Cl(a), then f(x) = f(y). The calculation which shows this is just the reversal of the steps above. To be precise: xax−1 = yay−1 ) ax−1y = x−1ya ) x−1y commutes with a ) x−1y 2 H ) xH = yH (Make sure you understand this last step.) QED Note that in a finite group this implies that jCl(a)j divides jGj. This observation lets us prove the following result: Theorem: If jGj = pm for p a positive prime, then Z(G) (the set of all elements that commute with all elements in the group) is non-trivial.
    [Show full text]
  • 2.2 the Centre, Centralizers and Conjugacy
    2.2 The centre, centralizers and conjugacy Definition 2.2.1. LetG be a group with operation�. The centre ofG, denoted byZ(G) is the subset ofG consisting of all those elements that commute with every element ofG, i.e. Z(G) ={x G:x�g=g�x for allg G}. ∈ ∈ Note that the centre ofG is equal toG if and only ifG is abelian. Example 2.2.2. What is the centre ofGL(n,Q), the group ofn n invertible matrices with rational × entries (under matrix multiplication)? Solution (Summary): Suppose thatA belongs to the centre of GL(n,Q) (soA is an n invertible × matrix). Fori,j in the range 1, . ,n withi=j, letE denote the matrix that has 1 in the(i,j) � ij position and zeros in all other positions. ThenI n +E ij GL(n,Q) and ∈ A(I +E ) = (I +E )A= A+AE =A+E A= AE =E A. n ij n ij ⇒ ij ij ⇒ ij ij NowAE ij has Columni ofA as itsjth column and is otherwise full of zeros, whileE ijA has Row j ofA as itsith row and is otherwise full of zeros. In order for these two matrices to be equal for alli andj, it must be that the off-diagonal entries ofA are all zero and that the entries on the main diagonal are all equal to each other. ThusA= aI n, for somea Q,a= 0. On the other hand it is ∈ � easily checked that any matrix of the form aIn wherea Q does commute with all other matrices.
    [Show full text]
  • Conjugacy Growth and Hyperbolicity
    Conjugacy growth and hyperbolicity Laura Ciobanu University of Neuch^atel,Switzerland Webinar, December 3, 2015 1 / 48 Summary 1. Conjugacy growth in groups 2. Conjugacy growth series in groups 3. Rivin's conjecture for hyperbolic groups 4. Conjugacy representatives in acylindrically hyperbolic groups 2 / 48 and by jgjc the conjugacy length of [g], where jgjc is the length of the shortest h 2 [g], with respect to X . I The conjugacy growth function is then σG;X (n) := ]f[g] 2 G j jgjc = ng: Counting conjugacy classes Let G be a group with finite generating set X . I Denote by [g] the conjugacy class of g 2 G 3 / 48 I The conjugacy growth function is then σG;X (n) := ]f[g] 2 G j jgjc = ng: Counting conjugacy classes Let G be a group with finite generating set X . I Denote by [g] the conjugacy class of g 2 G and by jgjc the conjugacy length of [g], where jgjc is the length of the shortest h 2 [g], with respect to X . 3 / 48 Counting conjugacy classes Let G be a group with finite generating set X . I Denote by [g] the conjugacy class of g 2 G and by jgjc the conjugacy length of [g], where jgjc is the length of the shortest h 2 [g], with respect to X . I The conjugacy growth function is then σG;X (n) := ]f[g] 2 G j jgjc = ng: 3 / 48 I Conjecture (Guba-Sapir): most (excluding the Osin or Ivanov type `monsters') groups of standard exponential growth should have exponential conjugacy growth.
    [Show full text]
  • A Note on Conjugacy Classes of Finite Groups
    Proc. Indian Acad. Sci. (Math. Sci.) Vol. 124, No. 1, February 2014, pp. 31–36. c Indian Academy of Sciences A note on conjugacy classes of finite groups HEMANT KALRA and DEEPAK GUMBER School of Mathematics and Computer Applications, Thapar University, Patiala 147 004, India E-mail: [email protected]; [email protected] MS received 29 August 2012; revised 27 November 2012 Abstract. Let G be a finite group and let xG denote the conjugacy class of an element x of G. We classify all finite groups G in the following three cases: (i) Each non-trivial conjugacy class of G together with the identity element 1 is a subgroup of G, (ii) union of any two distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G, and (iii) union of any three distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G. Keywords. Conjugacy class; rational group. 2000 Mathematics Subject Classification. 20E45, 20D20. 1. Introduction Throughout this paper, unless or otherwise stated, G denotes a finite group. The influence of arithmetic structure of conjugacy classes of G, like conjugacy class sizes, the number of conjugacy classes or the number of conjugacy class sizes, on the structure of G is an extensively studied question in group theory. Many authors have studied the influence of some other kind of behaviours of conjugacy classes on the structure of the group. For example, Dade and Yadav [2] have classified all finite groups in which the product of any two non-inverse conjugacy classes is again a conjugacy class.
    [Show full text]
  • If There Is F ∈ G Such That − G = F · H · F 1
    LECTURE 5 Conjugacy PAVEL RU˚ZIˇ CKAˇ Abstract. We study the conjugacy relation on a group. We prove that the conjugacy classes form a partition of the group. We show that a subgroup is normal if and only if it is a union of conjugacy classes. We will study some examples. we prove that permutations in a symmetric group are conjugated if and only if they have the same type. Finally we study sets of generators of a group. Understanding some sets of generators of an alternating group will help us to prove that all standard positions of the 15 puzzle corresponding to even permutations are solvable. 5.1. Conjugacy. Elements g, h of a group G are said to be conjugate (which we denote by g ∼ h) if there is f ∈ G such that − g = f · h · f 1. In this case we say the g is conjugate to h by f. Clearly g is conjugate to g by the unit of G and if g is conjugate to h by f then h is conjugate to g by f −1. It follows that the relation ∼ is reflexive and symmetric. If g is conjugate with h by f and h is conjugate with k by e, then g is conjugate with k by f · e, indeed, (f · e) · k · (f · e)−1 = f · e · k · e−1 · f −1 = f · h · f −1 = g. Thus we have the transitivity of ∼. We conclude that the conjugacy form an equivalence relation on G. The blocks of ∼ are called the conjugacy classe (of conjugacy blocks) of G.
    [Show full text]