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# Group Theory

### 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

### Group Theory :Homomorphism of a Group: Verify if the mapping defined is a homomorphism and in that case in which it is homomorphism,determine the Kernel:G is the group of non-zero real numbers under multiplication, ¯G = G, phi(x) = x^2 all x belongs to G.

Modern Algebra Group Theory (L) Homomorphism of a Group

### Group Theory : If H is a subgroup of G, let N(H) = {g belongs to G|gHg^-1 = H}. Prove that H is normal in N(H).

Modern Algebra Group Theory (XLVI) Normal Subgroups of a Group

### Normal Subgroups of a Group

Modern Algebra Group Theory (XLIV) Subgroups 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

### Group theory, cardinality and isomorphic example problems

I would like to know how to identify and prove the cardinality of sets and how to identify isomorphic. (See attached file for full problem description) Group Theory: a. If S and T are sets then let TS denote the set of all functions from S to T. Prove that the cardinality of TSxU equals the cardinality of (TS)U b. C

### 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 - Relation between Cyclic Group and Abelian Group : If the group G has five elements, show it must be abelian.

Modern Algebra Group Theory (XIX) Relation between Cyclic Group and Abelian Group If the group G has five elements, show it must be abeli

### 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

### Group Theory - Symmetric Set of Permutations

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

### 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.

### If is a group such that (ab)^2=a^2b^2 for all a,b belongs to G, show that G must be abelian. Or, Show that the group G is abelian iff (ab)^2=a^2b^2.

Modern Algebra Group Theory (VIII) If G is a group such that (ab)^2=a^2b^2 for all a,b belongs to G, show that G must be abelian. Or, Show that the group G is abelian iff (ab)^2=a^2b^2.

### 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

### Time complexity of an algorithm in theta notation

How much time does the following algorithm require as a function of n? Express your answer in "theta notation" in the simplest possible form. Show all work! l = 0 for i = 1 to n for j = 1 to i for k = j to n l = l +1

### Period of a Group of Elements

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.

### Cyclic Groups, Generators and Orders of Elements

(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.

### 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 & Quotient Group

See the attached file. 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) Dete

### Group Theory : Symmetry Groups, Cyclic Subgroups

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 a

### 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