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Group Theory

Group Theory refers to the study of algebraic structures known as groups. A group is a set of elements and an operation which can combine two elements (of the group) to form a third element. For example, consider the following equation:

s•x = t

s, x and t are all unknown variables. Group Theory is concerned with systems in which the above equation has a unique solution, which is the scenario when only one specific solution set exists. Group Theory does not concern itself with values, such as the specific value of s or t, but rather it deals with finding properties common to mathematical systems which obey a certain set of rules of Group Theory.

The basic rules of an algebraic group are as follows:

-closure: if a and b are in the group, then (s•t) is also in the group

-associativity: (s•t) •r = s•(t•r)

-identity: s•e=e•s = s

-inverses: s•s^(-1) = e

If a particular mathematical system obeys these rules, then the study of that system would form the foundation of Group Theory. Thus, understanding Group Theory is extremely important for the study of systems which are not only limited to Mathematics, but also extend into the study of Physics and Chemistry.

Abstract algebraic operations

Show that the operation * on the real number is defined by a*b = ab + a^2 is neither associative nor commutative. Show working --------------- Show that * on R is defined by a*b = a+b+2 is both associative and commutative Show working --------------- * The movements of a robot are restricted to no change (N), t

Facts About Group Theory

Please help with the following problem. Give 5 interesting facts about group theory (or tensor calculus). Each one need only be 1 to just a few sentences - please list separately and under each fact provide any links to websites or pictures etc. These facts need not be anything original. Thank you for your time and input.

Groups and Representations

Suppose that G is a finite group and H is a normal subgroup of G. (i) Explain briefly how the quotient group G/H is defined. [You should explain what the elements of G/H are, how the multiplication is defined, what the identity element is and how inverses are defined. You do not need to prove that the multiplication is well-

Rings, Quaternions and Dihedral Groups

1. Define the ring of quaternions H := { a1 + bi + cj + dk : a, b, c, d <- R }, with the relations 1 = 1 and i^2 = j^2 = k^2 = ijk = -1. Define the quaternion absolute value by |a1 + bi + cj + dk|^2 := a^2 + b^2 + c^2 + d^2 . Note H is actually isomorphic to R^4 as a vector space, but it has more structure than

Group and Ring Theory problems.

Group problems. 1. Let G be a group. Given a E G define the centralizer Z(a) := {b E G : ab = ba}. Prove that Z(a) (less than or equal to) G. For which a E G is Z(a) = G? 2. We say a, b E G are conjugate if there exists g E G such that a = gbg^-1. Recall (HW2.8) that this is an equivalence relation. Let C(a) := {b E G : E

Group Theory Proof: Pigeonhole Principle and Inverse Elements

Please help with the following problems involving group theory proofs. Provide step by step calculations along with explanations. a) Prove that if G is a finite group and a is an element of G then for some positive m , a^m is equal to the identity of G. (Use the Pigeon hole principle) b) Prove that if G is a finite group

Group theory

1. Let G=GL(2,Q), Q meaning rational numbers, and let A =matrix 0 -1 1 0 and B =matrix 0 1 -1 1 Show that A^4 = I = B^6, but that (AB)^n does not equal I for all n >0. Conclude that AB can have infinite order even though both factors A and B ha

Two questions in group theory

Problem #1 Prove that Aut(V)= (S3)and that Aut(S3)= S3. Problem #2 If H and K are normal subgroups of a group G with HK = G. Prove that G/(H n K) = (G/H) x (G/K).

Group Theory : Special Topics and Finite Simple Groups

Text Book: - Contemporary Abstract Algebra, Author:- Gallien In page number 430 & 431 , I need following questions to be answered 6, 7, 8, 16 & 17. Last Question:- Give examples of groups with specific properties (Simple group & Non Simple Groups) At least 10 examples. (Please refer Chapter 25).? Please mention each

Fibonacci Sequences

The Fibonacci Sequence is a recursively defined sequence determined by the function: Fn = 0 if n = 0 Fn = 1 if n = 1 Fn-2 + Fn-1 if n &#8805; 2 where n is a natural number. The first few terms of the sequence are: F0 = 0, F1 = 1, F2 = 1, F3 = 2, F4 = 3, F5 = 5, F6 = 8, F7 = 13, F8 =21 .....

Cyclic Groups and Generators

Let G = <a> be a cyclic group of size 600. What is a generator for the smallest subgroup of G that includes both a^42 and a^70?

Symmetric Groups

It is trivial that S[n] is cyclic for n = 1, 2, but is S[n] ever cyclic for n>=3? Prove why or why not.

Isomorphic : Noncyclic Group Order 4

Let V be a noncyclic group of order 4. (We know that all such groups are isomorphic, one is given in example 2.96). How large is Aut(V)? To which familiar group is Aut(V) isomorphic?

Cyclic Normal Subgroup

Let X be a prime. Prove or disprove that is cyclic for each normal subgroup K. See attached file for full problem description.

Direct Products of Groups

Let G=(x) x (y) where |x|=8 and |y|=4 a) Find all pairs a,b in G such that G=(a)x(b) (where a,b are expressed in terms of x and y) b) Let H = (x^2)x(y^2) be isomorphic to (Z/4 x Z/2). Prove that there are no elements a,b of G such that G=(a)x(b) and H=(a^2) x (b^2)

Solvable Groups and Chains of Subgroups

A) Prove that if H is nontrivial normal subgroup of the solvable group G then there is a nontrivial subgroup A of H with A normal subgroup of G and A abelian. b)Prove that if there exists a chain of subgroups G1<=G2<=.....<=G such that G=union(from i=1 to infinity)of Gi and each Gi is simple, then G is simple Part a of thi

Sylow p-Subgroups, Conjugacy and Abelian Groups

A) Let G be a group of order 203. Prove that if H is normal subgroup of order 7 in G then H<=Z(G). Deduce that G is abelian in this case. b)Let P be a normal Sylow p-subgroup of G and let H be any subgroup of G. Prove that P intersect H is the unique Sylow p-subgroup of H. c)Let P be in Syl_p(G) and assume N is a normal su

Group Action, Conjugates and Conjugation

Consider the group action of on itself via conjugation. ={ }, and a) Find all the elements of that are fixed by the element r. b) Let G be a group, and consider the action of G on itself via conjugation. Let g to G. Prove or disprove that the set of all elements of G that are fixed by g is a subgroup of G. c) Find a

Odd Order and Cyclic Groups

Suppose that G is a finite group of odd order 2n + 1. Prove or disprove that the number of cyclic subgroups of G is at most n + 1.

Group Homomorphisms and Isomorphisms

Let &#966;: G --> H be a group homomorphism. Let &#966; be a surjection. Then &#966; is an isomorphism if and only if the order of the element &#966;(a) is equal to the order of the element a for all a &#1028; G.

Dihedral Groups : Group Action

Consider the "usual action " of Dihedral group of order 10 (D10) on the set {1,2,3,4,5} and define D10 on the set of all 2-element subsets of {1,2,3,4,5} by g*{i,j} ={g*i,g*j} Find all the 2-element sets that are fixed by the element r i.e r*{i.j}={i,j} I think there would be nothing, but i am sure . So could someone gi

Groups

2.b) Consider G= , x*y be the fractional part of x+y .(i.e:x*y=x+y-[x+y] where [a] is the greatest integer less than or equal than a ) Construct a group H such that for each there exists an element of order , but non of the other orders are present.(Hint : use a subqroup of ): I want to claim the group is H= under ad