Indian Institute of Technology Guwahati
Tutorial Sheet-1
July-Nov 2015 semester
MA 521, Modern Algebra
Instructor: Shyamashree Upadhyay
Q. 1: (a) For n = 25 , find all positive integers less than n and relatively prime
to n.
(b) Determine 51 mod 13, (82.73) mod 7 and (51+68) mod 7.
Q 2: Show that if a and b are positive integers, then
ab= lcm (a, b) gcd (a, b).
Q 3: Let n be a fixed positive integer > 1 and let a and b be positive integers.
Prove that a mod n = b mod n if and only if a = b mod n.
Q 4: Let n and a be positive integers and let d = gcd (𝑎, n ). Show that the
equation 𝑎x = 1 mod n has a solution if and only if d= 1.
Q 5: Let S be the set of all real numbers. If 𝑎, b Є S, define a ~ b if a – b is an
integer. Show that ~ is an equivalence relation on S. Describe the equivalence
classes of S.
Q 6: Give two reasons why the set of all odd integers under addition is not a
group.
Q 7: Prove that a group G is abelian if and only if (ab) – 1 = a– 1 b– 1
Q. 8: Prove that if (ab)2 = a2 b2 in a group G, then ab = ba.
Q 9: Prove that the set { 1, 2, . . . , n – 1 } is a group under multiplication modulo
n if and only if n is a prime.
Q 10: If a group G has 3 elements, show that it must be abelian.
Q 11: If G is a finite group, show that there exists a positive integer N such
that aN = e ∀ a Є G.
𝑎
b
) where ad ≠ 0. Prove
0 d
that G forms a group under matrix multiplication. Is G abelian ?
Q 12: Let G be the set of all real 2 х 2 matrices (
Q 13: Show that if every element of a group G is its own inverse, then G is
abelian.
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