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Group Theory

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

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, 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

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.

Sylow's Theorem Group Theory

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.

Sylow's Theorem Symmetric Group

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.

Normal Subgroup of Order

Modern Algebra Group Theory (LXXV) Normal subgroup of a group The group of order p^2, where p is a prime number Prove that a group of orde

Nilpotent Groups

Let G = UT (n,F) be the set of the upper triangular n x n matrices with entries in a field F with p elements and 1's on the diagonal. The operation in G is matrix multiplication. (a) Show that G is a group (b) Show that G is a finite p-group (c) Consider the upper central series of G: 1 = Z_0 (G) <= Z_1 (G) <= Z_2 (G) <