Math 416 - Abstract Algebra

Chapter 5 - isomorphisms, cosets

Practice  questions

The following questions are based on sample questions from the "Instructors' Solutions Manual and on the true-false questions on Gallian's site

True-False

1.         S_n is non-abelian for all n > 3.

2.         If a is a permutation that is an m-cycle and b is a permutation that is an n-cycle, then |ab| = lcm(m,n).

3.         Every group is isomorphic to a group of permutations.

4.         Every infinite cyclic group is isomorphic to Z.

5.         Every cyclic group of order n is isomorphic to Z_n.

6.         D_n has a subgroup isomorphic to Z_n.

7.         A group can be isomorphic to a proper subgroup of itself.

8.         Two groups isomorphic to the same group are isomorphic to each other.

9.         If a finite group has order n, then the group contains a subgroup of order d for every positive divisor d of n.

10.       If a Î G, where G is a finite group, then |a| divides |G|.

11.       If H is a subgroup of G and if a Î G, then |aH| = |Ha|.

Other problems and questions

12.       Prove or disprove that U(15) is isomorphic to (24)

13.       Suppose that f is an automorphism of Z ₉ and that f(4) = 1.  Determine f y giving a formula, a rule, or a table.

14.       Let .  Prove that G under matrix multiplication is isomorphic to Z under addition.

15.       Prove that  is an automorphism from U(8) to U(8)

16        Prove that 11Z is isomorphic to Z.

17        Find all the left cosets of {1, 11} in U(20)

18.       Given that |a| = 20, find all the left cosets of < a¹² > in < a >

19        Suppose that K is a subgroup of H, and that H is a subgroup of G.  Suppose that there are six left cosets of K in H and there are four left cosets of H in G.  How many left cosets of K are there in G?

20.       If H and K are subgroups of G and if g Î G, prove that gH Ç gK = g (H Ç K).

21.       Prove that the only subgroup of D₅ that contains two reflections is D₅ itself.

22.       Prove that S₄ has a subgroup that is isomorphic to D₄.

23.       Prove that S₄ has no subgroup that is isomorphic to D₅.

24.       Prove that  is an isomorphism from R, + to R⁺, _°, (where the little circle indicates that your group operation is multiplication.)

25.       State and prove Lagrange’s theorem.

26.       I am thinking of a group, G, generated by two elements, x and y. 

In G, | x | = | xy | = 3, and | y | = 2.  (You may not assume that G is abelian.)

Find | G |.

The test will be strongly based on these questions.

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