please answer all questions in the pmath347 assignment1

 

 

PMATH 347 Assignment 1 Fall 2016 This assignment is due at the beginning of the class on Wednesday, September 21.
Question 1
Let 

1 2 3 4
σ=
2 4 1 3 

1 2 3 4
and Ï„ =
.
3 4 1 2 In each case, solve for χ ∈ S4 .
(a) χτ σ = ε.
(b) τ χσ −1 = σ.
Question 2
 Let S be a non-empty set and let M = X : X ⊆ S denote the set of all subsets of S. Let ∪ and
∩ denote the usual set union and set intersection.
(a) Determine if (M, ∪) is a monoid or not. Justify your answer.
(b) Determine if (M, ∩) is a monoid or not. Justify your answer.
(c) Determine if the empty set is a monoid. Justify your answer.
Question 3
Let G be a group and a, b ∈ G.
(a) Suppose that a6 = 1 and ab = ba2 . Prove that a3 = 1 and aba = b.
(b) Suppose that for n ∈ N ∪ {0}, we have (ab)n = 1. Prove that (ba)n = 1.
(c) Extend the statement in (b) to all n ∈ Z.
Question 4
(a) Prove that up to isomorphism, there are only two groups of order 4, the cyclic group C4 and a
noncyclic group K4 , whose Cayley table is shown below.
K4
1
a
b
c 1
1
a
b
c a
a
1
c
b b
b
c
1
a c
c
b
a
1 The group K4 is called the Klein 4 group.
(b) Prove that K4 ∼
= C2 × C2 .
Question 5
(a) Prove that a group G is abelian if and only if (gh)−1 = g −1 h−1 for all g, h ∈ G.
(b) Prove that a group G is abelian if g 2 = 1 for all g ∈ G.
(c) Determine if the converse of (2) is true or not. Justify your answer.

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