List all positive integers that can occur as the order of an element of G.

Algebra by Geoffrey
Math 100A (Greenfeld/Kedlaya) Midterm 1 – UC San Diego, October 14, 2022
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Instructions:
You have 50 minutes. No textbooks, notes, or outside resources are allowed. Use the space
provided for your answers. If you need more space, do not write on the back of the page; instead,
use the extra page at the end and indicate clearly which problem(s) you are answering there. Unless
otherwise specified, you must fully justify your answers.
Question Score Maximum
1 10
2 10
3 10
4 10
5 10
Total 50
Problem 1. [10 points.]

(i) State the cancellative law in a group G.
(ii) Give the definition of the group GL2(R).
(iii) Give the definition of an isomorphism φ : G → G′ of groups.

Problem 2. [10 points.]

Let G be a cyclic group of order 15.
(i) List all positive integers that can occur as the order of an element of G.
(ii) For each number you listed in (i), count the number of elements of G of that order. (Reminder:
these counts should add up to 15.)

Problem 3. [10 points.]
(i) Let φ : G → H be a homomorphism. Prove that if G is abelian, then both the kernel and the
image of φ are abelian.
(ii) Describe a homomorphism from S3 to a group of order 2 whose kernel and image are both
abelian, even though S3 is not itself abelian.

Problem 4. [10 points.]

(i) Compute the product (123)(345) in the group S5.
(ii) Compute the product (345)(123) in the group S5.
(iii) Verify that your answers to (i) and (ii) are conjugate elements of S5.

Problem 5 [corrected]. [10 points.]

Let a, b be elements in a group G. Assume that a5 = 1 and a3b = ba3. Prove that ab = ba.

Extra page.

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