### Group Theory - Symmetric Set of Permutations..

Modern Algebra Group Theory (XIV) Symmetric Set of Permutations Symmetric Set of Permutations : Find order of all elements in S_3, where S_3 is the symme

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Modern Algebra Group Theory (XIV) Symmetric Set of Permutations Symmetric Set of Permutations : Find order of all elements in S_3, where S_3 is the symme

Modern Algebra Group Theory (XI) Symmetric Set of Permutations In S3 give an example of two elements x,y such that (x.y)^2 is not equal to x^2.y

Group Theory (X) In a group G in which (a.b)^i =a^i.b^i for three consecutive integers for all a,b belongs to G, then G is abelian. Show that the conclusion does not follow if we assume the relation (a.b)^i =a^i.b^i for just two consecutive integers.

Let G be any non-Abelian group of order 6. By Cauchy's theorem, G has an element, a, of order 2. Let H = a, and let S be the set of left cosets of H. (a) Show H is not normal in G. (Hint: If H is normal, then H is a subset of Z(G), and then it can be shown that G is Abelian) (b) Use the below result and part (a) to show th

Let X be a path-connected space and suppose that every map f: S^1 --> X is homotopically trivial but not necessarily by a homotopy leaving the base point x_0 fixed. Show that pi_1(X,x_0) = 0.

(1) Let G be a group such that ... Show that G cannot be cyclic. (2) Show that a cyclic group with one generator has at most 2 elements. (3) Let a 2 G be an element of order two, and b 2 G an element of order three. Show that HK where H = (a) and K = (b) has order 6. See the attached file.

Let G = (Z/3Z)^4 SemiDirectProduct S_4 be the semi-direct product of (Z/3Z)^4 and S_4. Here S_4 acts on (Z/3Z)^4 by permutating the coordinates. Hint: Given H1, H2 an element in (Z/3Z)^4 and K1, K2 an element in S4. The semi-direct product is given by the operation (H1, K1) * (H2, K2) = (H1 + K1(H2), K1 * K2) A) Find the C

Thisquestion is concerned with subgroups ofthe group S5 of permutations on the set {1,2,3,4,5} , a group with 120 elements. (a) Explain why this group has cyclic subgroups of order 1,2,3,4,5 and 6, and give examples of each of these. Explain why this group does not have cyclic subgroups of any other order. (8 marks (b) By co

3 (a) (i) Let G=Z12(sub12 don't know how to put it), the group of integers modulo 12. Prove that H= {0, 6} AND K= {0, 4, 8} are subgroups of G. Calculate the subset H+K formed by adding together all possible pairs of elements from H and K, i.e. H+K= {h+kh is a subgroup of H, k is a subgroup of K} Prove that this is also a

(a) If G1 and G2 are groups, define what it means to describe a function h:G1 -> G2 as a homomorphism. (b) If h: G1 ?> G2 is a homomorphism, define the kernel of h.Prove that the range of h is a subgroup of G2 , and that the kernel of h is a normal subgroup of G1. (c) Let G be the group of 2x2 real matrices under additi

2.Let G be abelian of order n. If gcd(n;m) = 1, prove that f(g) = gm is an automorphism of G. (Note: Automorphism is just an isomorphism from G to itself.) 3. If f : Z7 ! Z5 is a homomorphism, prove that f(x) = 0 for all x 2 Z7. 4. Prove that in the group S10 every permutation of order 20 must be odd. 5. Suppose G is a group

1. let H be a subgroup of a group G such that g ֿ ¹ hg elements in H for all h elements in H. Show every left coset gH is the same as the right coset Hg. 2. prove that if G is an abelian group, written multiplicatively, with identity element e, then all elements, x, of G satisfying the equation x²=e form a sub group H o

? I have the following cayley tables (which is in modulo 9) determine the order of each element . Prove that G is a cyclic group. ? Let be the symmetric group of degree 3 together with composition of maps. Is G isomorphic to ? Justify your answer. ? Let p be a prime number and G a group of order with identity e

Find H K in {see attachment}, if H={[1],[8]}, and K = {[1], [4], [10], 13], [16],[19]}.

Let G sub1 and G sub 2 be groups, with subgroups H sub 1 and H sub 2, respectivetly. Show that {(x sub 1, x sub 2) | H sub 1 is an element of H sub 1, x sub 2 is an element of H sub 2} is a subgroup of the direct product G sub 1 x G sub 2.

Prove that if G1 and G2 are abelian groups, then the direct product G1 x G2 is abelian.

A number of questions involving rings and groups. Example: 3) Let R be a ring and [equationA]. Let [equationB] be the ring of n x n matrices with entries in R. What is the identity element of S? *(Please see attachment for complete list of problems)

Five Problems: Let G=[FORMULA1], with operation given by multiplication modulo 14. 1) by computing the Cayley table of G, or otherwise, show the G is a group. You may assume that without proof multiplication modulo 14 is associative. 2) Prove that the subset H={1,9,11} is a subgroup of G 3) Compute the left cosets of G a

1) Recall that So denotes the symmetric group of degree 6, the group of permutations f the numbers 1 to 6 ... see attached for full questions 1) Recall that denotes the symmetric group of degree 6, the group of permutations f the numbers 1 to 6. let . Thus is a bijection, mapping 1 to 3, 2 to 5, etc. let • C

1) Let G be the set of 2x2 matrices given by show that G is a group under matrix multiplication. You may assume that matrix multiplication is associative. 2) Let G be a group,... prove that G is Abelian. See attachment.

Note: G' means derived (commutator) subgroup of G and Sn is symmetric group of degree n Please find G' in each case (a) G is abelian (b) G = Sn

If K is a normal subgroup of G has index m, show that g^m belongs to K for all g belonging to G.

If G is any group, define $:G->G by $(g) = g^-1. Show that G is abelian if an only if $ is a homomorphism.

Show that every cyclic group Cn of order n is abelian. (Moreover, show that if G is a group, so is GxG)

What is the number of different conjugated classes of elements of order 2 in S_5.