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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.

### Let G be a group. Show that O(a^n) = O(a)/(O(a),n) for all a belongs to G, where n is an integer and (O(a), n) = greatest common divisor of O(a) and n.

Let G be a group. Show that O(a^n) = O(a)/(O(a),n) for all a belongs to G, where n is an integer and (O(a), n) = greatest common divisor of O(a) and n. The details are given in the attached file.

### 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

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.

### Proof in Group Theory

Let C* be the multiplicative group of nonzero complex numbers and R** be the multiplicative group of positive real numbers. Prove that C* is isomorphic to R** X R/Z where R is the additive group of real numbers.

### Problems in Group Theory.

Let i be an integer with 1 <= i <= n. Let Gi* be the subset of G1 X ... X Gn consisting of those elements whose ith coorinate is any element of Gi and whose other coordinates are each of the identity element, that is, Gi* = {(ei,...ei-1,ai,ei+1,...,en | ai in G} Show that Gi* is a normal subgroup of G1 X ... X Gn G

### Modular arithmetic and group theory

Show that the set {0,1,2,3} is not a group under multiplication modulo 4. The inverse property says that every element of the group has an inverse. When an element and its inverse are combined under and operation the result is the identity element. The identity element for multiplication is 1.

### Group Properties of Even and Odd Integers

(a) Verify that 2Z (the set of even integers) forms a group under ordinary addition. (b) Give two reasons why the set of odd integers would not form a group under ordinary addition. See Attachment.

### Problems in Group Theory

See the attached file. 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

### Problems in Group Theory: Homomorphism

Let G_1 and G_2 be groups and ?:G_1?G_2 a map. Which of the following is a group homomorphism? Explain your answers. If ? is a homomorphism, describe the kernel and the image of ?. a) G_1=C_4=?a|a^4=e?,G_2=Z_2 (the integers modulo 2 with the operation +), ?:a^i?i (mod 2). b) G_1=G_2=Z_5 (the integers modulo 5 with the op

### 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

### Abelian Subsets

Let G be a group and S any subset of G. Prove that C_G (S) = {g in G such that gs = sg for all s in S} is a subgroup of G. Prove that Z (G) (center of G) = C_G (S) is abelian and is a normal subgroup of G.

### Examining Subgroups

Let G be a group with A and B subgroups of G. Prove that the set AB = {ab | a is in A and b is in B} is a subgroup of G if and only if AB = BA ( ie, for any a in A, b in B, there exist elements a_1 in A, b_1 in B such that ab = b_1a_1, and there exist elements a_2 in A, b_2 in B such that ba = a_2b_2)

### Abstract Algebra: Symmetric Group Problem

a) Let G = S_4. What orders do the elements have? Give reasons and examples. b) Without listing them, how many subgroups does G have of order 3? Why? c) Using examples and/or theorems, argue that G has at least one subgroup of every order dividing |G|.

### Clarification of Sylow's Theorem

A) Prove there is no simple group of order 200. b) Assume that a group G has two Sylow p-subgroups K and H. Prove that K and H are isomorphic. c) Show that a group G of order 2p^n has proper normal subgroup, where p is odd prime number and n > 0.

### Abelian Group Problems

Let G be group of order 54 a. If G is abelian, what groups can it be up to isomorphisms b. If G is nonabelian and the order of G is 24 and G is isomorphic to H x Z_3, what are the possibilities for H up to isomorphism c. If p and q are distinct primes, how many abelian groups are there of order p (square) q(fourth)?

### Group theory

1. Prove that the subgroup of A4 generated by any element of order 2 and and any element of order 3 is all of A4. 2. prove that if x and y are distinct 3-cycles in S4 with x != y^-1 (x not equal to the inverse of y), then the subgroup of S4 generated by x and y is all of A4. Thanks. Any help is appreciated.

### The quaternion group

Let G be a group generated by elements a and b such that |a| = 4, b^2 = a^2, and ba = a^3 b. Show that G is a group of order 8 and that G is isomorphic to the quaterunion group Q = {1, i, -1, -i, j, k, -j, -k }.

### Cyclic group

Let G = <a> be cyclic group of order n. a) Prove that the cyclic subgroup generated by a^m is the same as the cyclic subgroup generated by a^d, where d = (m,n) b) Prove that a^m is a generator of G if and only if (m,n) = 1

### Group theory

A) Let G be a nonempty finite set equipped with an associative operation such that for all a,b,c,d in G: if ab = ac, then b = c and if bd = cd, then b = c. Prove that G is a group. b) Show that part (a) may be false if G is infinite

### Category of Nilpotent Groups

Prove that there cannot be a nilpotent group N generated by two elements with the property that every nilpotent group generated by two elements is a homomorphic image of N (i.e.: free objects do not always exist in the category of nilpotent groups).

### 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

### Nonisomorphic Abelian Groups

In each of parts (a) to (e) give the number of non-isomorphic abelian groups of the specified order - do not list the groups. (a) 100 (b) 576 (c) 1155 (d) 42875 (e) 2704. Prove that your answer is correct and list the groups in part (e).

### Groups of Isomorphic Rational Numbers

Give an example of groups H_i, K_j such that H_1xH_2 is isomorphic to K_1xK_2 and no H_i is isomorphic to any K_j. Let G be the additive group Q of rational numbers. Show that G is not the internal direct product of any two of its proper subgroups. If G is the internal direct product of subgroups G_1 and G_2 show that G/G_

### Groups and matrices..

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

### Group Theory Elements

Let G be a group, let a, b be elements in G and let m and n be (not necessarily positive) integers. Prove that: i) (a^n)^m= (a^mn) ii) if a and b commute, then (ab)^n = a^n times b^n iii) a^m times a^n = (a)^(m+n)

### Group Theory

Let G be a finite group with K is a normal subgroup of G. If (l K l, [G:K]) =1, prove that K is unique subgroup of G having order l K l. I am having trouble with this proof and I need it written in complete proof form and I don't really know how to do it.

### Set Operations: Committee form from two groups of people

I have two committees A and B, with |A| = 3 = |B|, out of a group of 8 people such that at most one person can serve on both committees. How many ways we can form the committees A and B? Please show me how it's done.

### 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, Computing number of odd order subgroups

Question: Let G be a group of order 2k , that contains a cyclic subgroup of order k, where k is odd. Determine a formula to compute the number of subgroups of G that are of odd order.