Solutions to Problem Set 1

Math 345/504

1-26-2018

1. (a) Write the group element using additive notation. (Assume the operation is commutative.)

(b) Write the group element using multiplicative notation. (Assume the operation is commutative.)

(a) In additive notation, this is .

(b) In multiplicative notation, this is .

2. Let G be a group (with the operation written multiplicatively), and let a and b be arbitrary elements of G.

(a) Simplify as much as possible.

(b) Solve for x in terms of a and b, simplifying your answer as much as possible:

(a)

(b)

3. This is the operation table for a group G, written using multiplicative notation. The identity is 1.

(a) Compute .

(b) Compute .

(c) Compute .

(d) Give a specific example to prove that G is not abelian.

(a) .

(b) .

(c) .

(d) , but , so .

4. The following set is a group under multiplication mod 20:

So, for example, .

(a) Compute in G.

(b) Find in G.

(c) When the operation in a group G is written using multiplicative notation, the order of an element is the smallest positive power n such that . The identity 1 is the only element with order 1.

For example, , so the order of 11 is 2.

Find the order of 13 in G.

[Compute powers of 13, stopping at the first power of 13 which is equal to 1.]

(a) .

(b) Since , I have .

(c)

The order of 13 is 4.

5. The following set is a group under addition mod 12:

For example,

(a) Compute in .

(b) Find -8 in .

(c) When the operation in a group G is written using additive notation, the order of an element is the smallest positive multiple n such that . The identity 0 is the only element with order 1.

(Remember that " " is shorthand for . It is not multiplication in the group, since the operation is addition.)

For example,

Hence, the order of 4 is 3.

Find the order of 8 in .

[Compute multiples of 8 and stop at the first multiple which equals 0.]

(a) in .

(b) , so in .

(c)

Hence, the order of 8 is 3.

[Math 504]

6. Let G be a group. Prove that G is abelian if and only if for all .

Suppose G is abelian. Let . Then

Conversely, suppose that for all . Let . I must show that . I have

Therefore, G is abelian.

Courage consists of the power of self-recovery. - Ralph Waldo Emerson

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