National Institute of Technology Tiruchirappalli
Department of Mathematics
Algebra (MA702)
I M.Sc Mathematics Semester-II January 2020 Session
List of Problems for Assignment-1
1. If a group of order 52.7.11 has more than one Sylow 5-subgroup, exactly how many does it
have?
2. Prove that a group of order 105 contains a subgroup of order 35.
3. Suppose that Gis an Abelian group of order 120 and that Ghas exactly three elements of order
2. Determine the isomorphism class of G.
4. Suppose that the order of some finite Abelian group is divisible by 10. Prove that the group has
a cyclic subgroup of order 10.
5. The set {1,9,16,22,29,53,74,79,81}is a group under multiplication modulo 91. Determine the
isomorphism class of this group.
6. Suppose that aand bbelong to a commutative ring Rwith unity. If ais a unit of Rand b2= 0,
show that a+bis a unit of R.
7. Suppose that Ris a ring and that a2=afor all ain R. Show that Ris commutative.
8. Find all units, zero-divisors, idempotents, and nilpotent elements in Z3โZ6.
9. Find a subring of ZโZthat is not an ideal of ZโZ.
10. If nis an integer greater than 1, show that hni=nZ is a prime ideal of Zif and only if nis prime.
11. If an ideal Iof a ring Rcontains a unit, show that I=R.
12. How many elements are in Z[i]/h3 + ii? Give reasons for your answer.
13. In ZโZ, let I={(a, 0) |aโZ}. Show that Iis a prime ideal but not a maximal ideal.
14. Show that A={(3x, y)|x, y โZ}is a maximal ideal of ZโZ.
15. Show that Z[i]/h1โiiis a field. How many elements does this field have?
