### 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?

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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?

List all of the elements of order 15 in Z_600.

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. 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

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.

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

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.

(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?

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.

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

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)

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

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

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

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

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.

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 φ(a) is equal to the order of the element a for all a єG. See the attached file.

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

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.

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

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

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

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.

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 of order 9 or 27.

See the attached file. Modern Algebra Group Theory (CXII) Groups of Order Power of a Prime Another Counting Principle

See the attached file. Modern Algebra Group Theory (CXI) Permutation Groups Another Counting Principle If O(G) = pn a prime

Modern Algebra Group Theory (CX) Sylow's Theorem Find all 3-Sylow subgroups of or, Sylow 3-subgroups and 2-Sylow subgroup or, Sylow 2-subgroups of the symmetric group of degree 4, S4.

Modern Algebra Group Theory (CIX) Sylow's Theorem In the symmetric group of degree 4, S4 , find a 2-Sylow subgroup and a 3-Sylow subgroup.

Using O(N) = Σca for some choices of a in N , prove that in A5 there is no normal subgroup N other than (e) and A5 .