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Group Theory

Continuous Maps, Homomorphisms and Cyclic Groups

Let f: S^n --> S^n be a continuous map. Consider the induced homomorphism f*: H~_n (S^n) --> H~_n (S^n), where H~_n is a reduced homology group. Then from the fact that H~_n (S^n) is an infinite cyclic group, it follows that there is a unique integer d such that f*(u) = du for any u in H~_n (S^n). How exactly does "ther

Nilpotent Groups

Let G = UT (n,F) be the set of the upper triangular n x n matrices with entries in a field F with p elements and 1's on the diagonal. The operation in G is matrix multiplication. (a) Show that G is a group (b) Show that G is a finite p-group (c) Consider the upper central series of G: 1 = Z_0 (G) <= Z_1 (G) <= Z_2 (G) <

Group of order 9

This is the question: Consider small groups. (i) Show that a group of order 9 is isomorphic to Z9 or Z3 x Z3 (ii) List all groups of order at most 10 (up to isomorphism)

Symmetric groups

Symmetric groups: G = Sn. (i) Let g1, g2 belong G be two disjoint cycles, and let g = g1g2. Prove that o(g) = lcm { o( g1), o(g2)}, where lcm stands for the least common multiple. (ii) Let g= g1g2 ... gr belong G, where g1,g2, ... gr are disjoint cycles. Prove that o(g) = lcm {o(g1), o(g2), ... o(gr)}. Can you

Path-connected Space : Abelian Group

Let x0 and x1 be points of the path-connected space X. Show that Pi_1(X,x0) is abelian iff for every pair a and b of paths from x0 to x1, we have a'=b', where a'([f])=[a-]*[f]*[a];( a- means the reverse of a.) and [f] belongs to Pi_1(X,x0). a':Pi_1(X,x0)->Pi_1(X,x1).

Group Structure, Order of two groups

I have questions about constructing a group structure, how to identify the order of a paired group when they have different orders and method of figuring out the group identity and the inverse of a pair that contained in the paired group. --- If G and H are groups then explain how to equip G x H with a group structure. If G

Formation of a Group under an Associative Product

Question: Let G be a nonempty set closed under an associative product, which in addition satisfies: (a) There exists an e in G such that e.a = a for all a in G. (b) Given a in G, there exists an element y(a) in G such that y(a).a = e. Then G is a group under this product.

Group Theory - Group of Even Order

Modern Algebra Group Theory (XXII) Group of Even Order If G is a group of even order, prove it has an element a which is not equal to e satisfying a^2 = e. The fully formatted problem is in the attached file.

How do you show that G is not an abelian?

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.

A problem related to period(order) of an element in a group

Let a and x be elements in a group G. Prove that a and axb ,where b is the inverse of a, have the same period. Let G be a multiplicative group and a, x € G. Prove that for all n € N , (xax-1) = xanx-1 ( N is the set of natural numbers) Deduce that xax-1 has the same period as a

Group theory proofs

A. Let =2 +1 (2 (Power 2(power n))) Plus 1. Prove that P is a prime Dividing , then the smallest m such that P (2 -1) is m = 2 (hint use the Division Algorithm and Binomial Theorem) Please see attached.

Semi-Direct Products, S4 Groups and Homomorphisms

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

Group Theory : Conjugacy, Cayley Table, Subgroups and Quotient Groups

Define the notion of conjugacy as it applies in a general group. Prove that the inverses of a pair of conjugate elements are also conjugate. Prove that conjugate elements have the same order. (6 marks) The remainder of this question concerns the group G , whose Cayley table is as [TABLE] (b) Determine the inverse and th

Group Theory : Symmetry Groups, Cyclic Subgroups and Permutations

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

Group Theory : Abelian Groups and Subgroups

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

Group Theory : Homomorphisms, Kernels, Isomorphisms and Fields

(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

Group Theory : Homomorphism, Subgroups, Abelian Groups and Group Order

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

Group Theory 1. i. State the axioms for an equivalence relation ii. The relation n mod 3 divides the non-negative integers (i.e, n in Z such that n &#8805; 0) into how many partitions? Show that n = 0 mod 3 is an equivalence relation. 2. Prove that, for any matrices, A, B and C: A+B=B+A And: A+(B+C)=(A+B)+C ( i.e., that the matrix addition is both commutative and associative) For simplicity, prove these properties using 2x2 matrices. 3. Prove that addition modulo n, written + is: i. Associative. ii. Commutative. There are two ways to prove these properties. Each way requires a definition or two: i. For n &#8805; 2, 0 &#8804; a, b &#8804; n+1, a+ b= a+b if a+b< n a+n-n if a+b&#8805; n ii. Writing a for a mod n and (a+b) = a+ b, then: (p+ q) &#8801; (p +q ) Do the proof using both methods. Which is more "algebraic" (in the sense of "abstract" algebra)? 4. Prove that addition modulo n, written + is: i. Associative ii Commutative. ( extra definations required : a for a mod n and (pà?q) = pà? q, so (pà? q) (p à?q ) 5. i. State the axioms defining a group - If (Z, +) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z , identify the inverse. Also show that + is associative. - If (Z, à?) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z, identify the inverse. Also show that + is associative. - If (R, à?) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z , identify the inverse. Also show that + is associative. iii. In each case, deterimine whether the algebraic structure is a group. For each such group: - show how it satifies the group axioms - Draw the cayley table for the group and list the inverse elements i. For S=(0,2), a+2b &#8801; (a+b) mod 2 and aà?2b &#8801; (a à?b)mod2 a. (S,+ ) ( possibly an additive group) b. (S,&#8729; 2) (possibly a multiplicative group). ii. For S = (0,1,2) where n=2,3 and + and à? are defined as in the last part. a. (S,+n) ( possibly an additive group) b. (S,&#8729; n) (possibly a multiplicative group). Determine whether any of the groups is an abelian group. If any of them are abelian: i. state the conditions under which a group is abelian ii. show that the group is abelian 6. there are only two groups of order four (Z 4and v). How many groups are there of the order five? Draw cayley tables for each one of them( the element should be named a,b,c,d,e) is either (or both) of the groups of order four subgroup of any of those of order five? If so which one 7. for each of the following structures, state whethere it is a group. If it is, state whether it is abelian or not. i.For any set, A, the set of one-to one and onto functions, f: A &#8594;A under composition ( written "&#9702;"). ii.The set of all subsets of the three-element set (a,b,c) ( there are eight such subsets) under: a. Union b. Intersection iii. The set G=(a+b&#8730;5| a,b in Q) under addition and multiplication iv The set consisting of non-zero numbers under a. addition b. division v. The set (1,5,7,11) under multiplication modulo 12. Draw cayley table vi. The set (4,6) under multiplication modulo 12. draw cayley table vii. The set of real numbers under à?, where aà?b = 2(a+b) viii The set of real numbers under +, where a+b = a+b-10 ix. The set of rotational symmetries of a regular hexagon under composition x The following sets of permutations under composition i. (e,(12),(123),(1234)) ii. (e,(12), (34), (12), (34)) 8.Let G be a group, (G, * ) in which there is an element, a , such that g * g=g . prove that g=e 9.Prove that for every element, a, of a group, G, the order of a and a^-1 are the same ( including the case of an infinite order) 10.Let x and y be elements of a group, G. Prove that the elements xy and yx have the same orders 11.Find the subgroups of i. Z7 ii. Z8 iii Z9 12. i. determine which of the folowign are subgroups of under + a. (0) b. (-1,0,1) c. (n| n=10m for some integer m d.(p| p is a prime number e. (0,1,2,3,4) under addition modulo 5 ii. Determine which of the following are subgroups of under mulitiplication: a. (1, -1) b. (x |x=3, for some integer n c. (x |x=p/2&#8319; for some integers, p,n) d. (x| x=k 3 for some interger k

Group Theory Group Theory 1. i. State the axioms for an equivalence relation

Group theory : explained

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