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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

Topological Groups : Quotient and Open Maps

Let be p:G to G/N a quotient map.Is it open? Let be f:G to H an open map. Is it quotient map? Here G and H are topological groups, and N is an subgroup.

Group Theory : Permutation Groups

Prove that there is no a such that a^( - 1 ) ( 1 , 2 , 3 ) a = ( 1 , 3 )( 5 , 7 , 8 )

Group Theory :The order of an n-cycle, the order of the product of the disjoint cycles and order of a given permutation.

Modern Algebra Group Theory (LXXXIV) Permutation Groups

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

Group Theory : Let G be a group and Z(G), the centre of G, then G/Z(G) is equivalent to I(G) , where I(G) is the set of all inner automorphisms of G.

Modern Algebra Group Theory (LXXII) The Set of all Automorphisms of a Group The Set of all Inner Automorphisms of a Group

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).

Aut(G), the set of automorphisms of G, is also a group.

Modern Algebra Group Theory (LXX) The Set of all Automorphisms of a Group If G is a group, then Aut(G),

The Group Theory Concept: The Abelian Group

Please address the following problem: A group of order p^2,where p is a prime number, is abelian.

Group Theory : Verify if the mapping defined is a homomorphism and in that case in which it is homomorphism,determine the Kernel : G is any abelian group and ¯G = G, phi(x) = x^5 all x belongs to G.

Modern Algebra Group Theory (LIV) Homomorphism of a Group

Normal Subgroups of a Group

Modern Algebra Group Theory (XLI) Subgroups of a Group

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.

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

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.

Structure of Groups : Cauchy's Theorem, Order, Abelian Groups, Non-Abelian Groups, Isomorphisms and Subgroups

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

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

Fundamental Groups, Path-Connected Space, Connectivity and Homotopy

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.

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