Abstract algebra

jimi.docx

MATH 402

Quiz 1 Page 2

1) Give an example of functions f and g such that

(i) f is 1-1 (f: AB)

(ii) g is onto (g: BC), and

(iii) the composition fog is not onto.

Be sure to specify the domain and range and the rule for each function.

2) Choose a group H and clearly describe H and list the elements.

(i) Give an example of two elements of the group H which commute (i.e., ab=ba).

(ii) Give an example of two elements of the group H that do not commute.

3) Give an example of a group G and group H where (i) a G and G=and (ii) a H and H ≠.

4) Find gcd(600,425) using the Euclidean algorithm. Find s, t such that

gcd(600, 450)=600s + 450t.

5) List all elements of U(12). Find multiplicative inverse of every element in U(12). Show work.

6) Find A-1 where A = in M2 (19).

7) Let H be a group. Suppose g2 = e for all gH. Prove that H is Abelian.

8) Let H be a group, let G and K be subgroups of H, and suppose H=G K. Prove that either H=G or K=H.

9) Prove that U(12) is a cyclic, find all the subgroups of U(12), and list all the generators of U(12).

10) Complete the Cayley table for a group of order 6 generated by a and b where |b|=3 and |a|= 2 and ab=b2a.

(i) Express each group element in the form of anbm or bman where n, m ≥0.

(ii) Is the group abelian? Why, or why not? (iii) Find all the cyclic subgroups?

11)Let . Prove that G is a cyclic subroup of GL (2,).

Hint:

12) Show a proof by a counterexample. Find integers a, b, and c such that a|c, b|c, but abc.

13) Determine (and list) the subgroups of D6 of order 4 and cyclic subgroups of D6 of order 4.

14) Using the induction method, show that for all

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