### Ask Your question

Need a fast expert's response?

Submit orderand get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

### Search & Filtering

If σ is an even permutation, then σ ² = I . Write given statement is true or false, justify your answer.

Let S be a nonempty subset of plane 2 \ , it is known that every point ( , ) x y in S

satisfies “if x > 0 , then y > 0 ”. Consider the following properties possibly satisfied

by points( , ) x y in S :

(I) If x ≤ 0 , then y ≤ 0 .

(II) If y ≤ 0 , then x ≤ 0 .

(III) If y > 0 , then x > 0 .

Which of the above properties will have to be satisfied by all points( , ) x y in S?

(a) (II) only

(b) (III) only

(c) (I) and (II)

(d) (I) and (III)

(e) (II) and (III)

satisfies “if x > 0 , then y > 0 ”. Consider the following properties possibly satisfied

by points( , ) x y in S :

(I) If x ≤ 0 , then y ≤ 0 .

(II) If y ≤ 0 , then x ≤ 0 .

(III) If y > 0 , then x > 0 .

Which of the above properties will have to be satisfied by all points( , ) x y in S?

(a) (II) only

(b) (III) only

(c) (I) and (II)

(d) (I) and (III)

(e) (II) and (III)

1. Write a complete Cayley Table for D6, the dihedral group of order 6.

2. Prove that if G is a group with property that the square of every element is the identity, then G is

abelian.

3. Construct the Cayley table for the group generated by g and h, where g and h satisfy the relations

g

3 = h

2 = e and gh = hg2

.

4. Let H and K be subgroups of a group G such that gcd(|H|, |K|) = 1. Apply Lagrange’s theorem to

show that |H ∩ K| = 1.

5. Consider the group Z12 and the subgroup H =< [4] >= {[0], [4], [8]}. Are the following pairs of elements

related under ∼H? Justify your answer.

(a) [3], [11],

(b) [3], [7],

(c) [5], [11],

(d) [6], [9],

(e) find all left cosets of H in G. Are they different from the right cosets?

2. Prove that if G is a group with property that the square of every element is the identity, then G is

abelian.

3. Construct the Cayley table for the group generated by g and h, where g and h satisfy the relations

g

3 = h

2 = e and gh = hg2

.

4. Let H and K be subgroups of a group G such that gcd(|H|, |K|) = 1. Apply Lagrange’s theorem to

show that |H ∩ K| = 1.

5. Consider the group Z12 and the subgroup H =< [4] >= {[0], [4], [8]}. Are the following pairs of elements

related under ∼H? Justify your answer.

(a) [3], [11],

(b) [3], [7],

(c) [5], [11],

(d) [6], [9],

(e) find all left cosets of H in G. Are they different from the right cosets?

if H and G are two isomorphic groups then prove that Atu(H) is isomorphic to Atu(G).

The set of discontinuous functions from [1,0] to R form a ring with respect to pointwise addition and multiplication. Is this statement true or false, justify your answer.

The characteristic of a finite field is zero. Is this statement true or false, justify your answer.

If a ring has a unit, then it has only one unit. Is this statement true or false ,justify your answer.

If σ ∈ Sn (n ≥ 3 ) is a product of an even number of disjoint cycles, then sign (σ) = 1. Is this statement true or false, justify your answer.

If R is a ring and I is an ideal of R , then xr = rx ∀ x ∈ I and r ∈ R . Is this statement true or false, justify your answer .

Every element of Sn has order at most n . Is this statement true or false, justify your answer.