Math 720

Fall 2000

Homework 1

This first problem will serve to classify all cyclic groups.

1. Let G be a group. Show that the following conditions are equivalent.

I. G is cyclic.

II. G is the homomorphic image of Z.

III. All homomorphic images of G are cyclic.

IV. All subgroups of G are cyclic.

V. G is isomorphic to Z or Z_n.

2. Give an example of an uncountable cyclic group or show why one cannot exist.

3. Let G be cyclic of order n and let k be (another, but not necessarily different) integer.

a) If k divides n, then show that there is a unique subgroup of G of order k.

b) If k does not divide n, then show there is no subgroup of G of order k.

4. Let G be a group and Prove the following:

a)

b)

c)

5. Let be a group homomorphism and let

a) If , then either divides or

b) If is one to one, then

6. Compute Aut(Z₄) and Aut(Z₂Z₂).

7. Let S be a semigroup.

a) Show that S is a group if and only if both of the following two conditions hold:

I. There is an element such that (Left identity)

II. For all , there is an element such that (Left inverse)

b) Part a) shows that to have a group, all you really need assume is a semigroup with a left identity and left inverses. Is this result true if we replace one of the “lefts” with a “right”? That is, is a semigroup with a left identity and right inverses necessarily a group? Prove or give a counterexample.

8. Show that a group G is finite if and only if it has only finitely many subgroups.