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# Group Theory

### Groups of Order 18

Construct two non-isomorphic, nonabelian groups of order 18.

### Fibonacci Sequences: Prove the Formula

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 ≥ 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 .....

### Equivalent Definition of Equivalence Relation on a Group

Show that the following are equivalent: (a) ~ is an equivalence relation on a group G (b) ~ is reflexive and, for all elements a, b, c of G: if a ~ b and b ~ c, then c ~ a. See the attached file.

### Groups and Subgroups

Let G be a group and let H and K be subgroups of G. Prove that H&#8746;K is a subgroup of G if and only if H &#8838; K or K &#8838; H.

### A space whose fundamental group is non-abelian.

Give an example of a space whose fundamental group is non-abelian.

### 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?

### Group Elements : List all of the elements of order 15 in Z_600

List all of the elements of order 15 in Z_600.

### Center of a Group and Abelian Groups

In any group G, some of the elements of G commute with all of the other elements in G. The set of all such elements in G is called the centre of G, and is denoted by Z(G). Hence Z(G) = {g|xg = gx x is a subset of G}. For instance in any group the identity commutes with every element - so Z(G) is never empty. It should

### Group Theory Questions

Group Theory Questions. See attached file for full problem description. ( p is odd prime and a an integer not divisible by p ) the LEGENDRE SYMBOL IS DEFINRD BY = 1 if a is a quadratic residue of P, = -1 if a is a quadratic nonresidue of P. a. Evaluate b. Evaluate Hint 4567 is a prime. c. Does 17 ( m

### 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.

### Sylow P-Subgroups, Isomorphisms and Solvable Groups

1. a) If M and N are normal subgroups of G then G/M is isomorphic to a subgroup of G/M x G/N. b) If G/M and G/N are solvable, then G/(M intersect N) is solvable. 2. Let G be finite and P be a Sylow p subgroup of G. Suppose the normalizer of P in G is a subset of H is a subset of G. Show that the normalizer of H in

### 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? See the attached file.

### Abelian Groups, Z-Modules and Q-Modules

(6) If M is a finite abelian group then M is naturally a Z-module. Can this action be extended to make M into a Q-module?

### Groups : Identities and Inverses

If G is a group, then (1) the identity element of G is unique, (2) every a belongs to G has a unique inverse in G.

### 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

### Center of a Heisenberg Group

Let H(F) be the Heisenberg group over the field F. Determine which matrices lies in the center of H(F)

### 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 Homomrphisms and Isomorphisms

Let Φ: G --> H be a group homomorphism. Let Φ be a surjection. Then Φ 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 єG. See the attached file.

### 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

### Cyclic Groups

A) let a be the m-cycle (123.....m). how to show that a^i is also an m-cycle if and only if i is relatively prime to m. Here a is an element of group G that generates the m-cycle. b) How to prove that the order of an element in Sn equals the least common multiple of the lengths of the cycles in its cycle decomposition.

### Groups and Fractional Elements

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 addi

### Groups, Order and Commutativity

1. If x is an element of a group and x is of order n then the elements 1, x, x^2,...x^n-1 are distinct (don't know how to show this!) 2. Let Y=<u,v/u^4=v^3=1, uv=v^2u^2> Y here is a group show a) v^2=v^-1 b) v commutes with u^3 c) v commutes with u d)uv=1 e)u=1, deduce that v=1 and conclude that Y=1

### Dihedral Groups

Use the gemerators and relations for D(sub2n)=<r,s/r^n=s^2=1,rs=sr^-1> to show that if x is any element of D2n, that is not a power of r, then rx=xr^-1. Here D2n is the dihedral group of order 2n

### Groups and Distinct Elements

Let G be a group and let be two distinct elements. Let n be the order of g and m be the order of h, and suppose that n and m are relatively prime. Prove or disprove that there is no pair i,j , such that . Please see the attached file for the fully formatted problems.

### Groups and Distinct Elements

Let G be a group and let be two distinct elements. Let n be the order of g and m be the order of h. Prove or disprove that there is no pair i,j , such that . Please see the attached file for the fully formatted problems.

### Prove that a group of order 108 must have a normal subgroup

Prove that a group of order 108 must have a normal subgroup of order 9 or 27.