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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)
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?
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.