Proofs in Group Theory : Cayley Table, Subsets and Cosets - Five Problems

Five Problems: Let G=[FORMULA1], with operation given by multiplication modulo 14. 1) by computing the Cayley table of G, or otherwise, show the G is a group. You may assume that without proof multiplication modulo 14 is associative. 2) Prove that the subset H={1,9,11} is a subgroup of G 3) Compute the left cosets of G a ...continues

Herstein Problem : Sylow Subgroups

Find the possible number of 11-Sylow subgroups, 7-Sylow subgroups, and 5-Sylow subgroups in a group of order 5^(2)*7*11 ( 5 squared times 7 times 11) This is problem #21 pg 103 of Topics in Algebra.

Hernstein Problem : Sylow Subgroups

If G is a group of order 385 show that its 11-Sylow subgroup is normal and its 7-Sylow subgroup is in the center of G. From problem #9 pg 102 Topics in Algebra.

Hernsteins Problem : Sylow Theorem

If G is of order 108 show that G has a normal subgroup of order 3^k ( 3 to the k power), where k is greater than or equal to 2. On pg 102 #10 in Topics of Algebra.

Herstein Problem : p-Sylow Subgroups

Let G be a group of n x n matrices over the integers modulo p, p a prime, which are invertable. Find a p-Sylow subgroup of G. Topics of Algebra pg103 #20.

Symmetry : Symmetries of a Square

Please see the attached file for the fully formatted problems. • Let S be a square, with vertices labelled (anticlockwise), 1,2,3,4. a symmetry of S is a rotation or reflection which preserves the square (although it may change the position of the vertices). Note that a symmetry is determined by its effect on the vertices. ...continues

Homomorphisms

• Let G be a group and let a,b be two elements of G. The conjugate of b by a is, by definition, the element . The centralizer of a, denoted by s the set of all elements g in G such that ga=ag. i) Find all possible conjugates f the permutation ii) Find the centralizer p in . iii) Prove that for any element a in a g ...continues

Rings and Groups (Ring Theory, Quaternions, Homomorphisms, Matrices)

A number of questions involving rings and groups. Example: 3) Let R be a ring and [equationA]. Let [equationB] be the ring of n x n matrices with entries in R. What is the identity element of S? *(Please see attachment for complete list of problems)

Rings and Subrings

Please see the attached file for the fully formatted problems. 1. Ler R be a ring, and , prove, using axioms for a ring, the following • The identity element of R s unique • That –r is the unique element of R such tht (-r)+r = 0. (hint, for part 1, suppose that 1 and 1’ ate two identities of R, show that 1-1’ must be ...continues

Groups : Third Sylow Theorem

Suppose a simple group G of order 660 is a subgroup of the symmetric group S11 ( here S is with 11 as a subscript ) and x = {1,2,3,4,5,6,7,8,9,10,11}( a permutation like 1 goes to 2 and so on, I think ) in G. If P equals the span of x ( or ) determine the permutations which generate NGP ( here G is written as the subscript o ...continues