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Sample Midterm Exam for Abstract Algebra (MATH 3030) - Fall 2012, Study notes of Abstract Algebra

University of Mustansiriyah (UM)Abstract Algebra

A sample midterm exam for the abstract algebra course (math 3030) taught by toby kenney during the fall 2012 semester. The exam covers various topics related to groups, subgroups, permutations, and homomorphisms. Students are expected to solve basic and theoretical questions within the given time frame.

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2021/2022

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MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Sample Midterm
This sample midterm is deliberately longer than the actually midterm, to
better cover the range of possible questions that could be asked. I have provided
estimated times for each question. Based on similar estimated times, it should
be possible to complete the midterm examination in 40 minutes (out of the 50
available).
Basic Questions
1. Are the following multiplication tables groups? Justify your answers. [5
mins]
(a)
a b c
a c a c
b a b a
c c a c
(b)
a b c d e
a a b c d e
b b a e c d
c c d a e b
d d e b a c
e e c d b a
(c)
a b c d
a a b c d
b b a d c
c c d a b
d d c b a
2. Which of the following are groups: [6 mins]
(a) N={n∈Z|n>0}with the operation a∗bgiven by addition without
carrying, that is, write aand b(in decimal, including any leading zeros
necessary) and in each position add the numbers modulo 10, so for example
2456 ∗824 = 2270.
(b) The set of functions f:R→Rsuch that f(1) = 0 with pointwise
addition (i. e. (f+g)(x) = f(x) + g(x)).
(c) The set of real numbers with the operation x∗y=xy
x+y.
3. How many generators are there in the cyclic group Z28? [2 mins]
1
pf3
pf4

Partial preview of the text

Download Sample Midterm Exam for Abstract Algebra (MATH 3030) - Fall 2012 and more Study notes Abstract Algebra in PDF only on Docsity!

MATH 3030, Abstract Algebra

FALL 2012

Toby Kenney

Sample Midterm

This sample midterm is deliberately longer than the actually midterm, to better cover the range of possible questions that could be asked. I have provided estimated times for each question. Based on similar estimated times, it should be possible to complete the midterm examination in 40 minutes (out of the 50 available).

Basic Questions

  1. Are the following multiplication tables groups? Justify your answers. [ mins]

(a)

a b c a c a c b a b a c c a c

(b)

a b c d e a a b c d e b b a e c d c c d a e b d d e b a c e e c d b a

(c)

a b c d a a b c d b b a d c c c d a b d d c b a

  1. Which of the following are groups: [6 mins] (a) N = {n ∈ Z|n > 0 } with the operation a ∗ b given by addition without carrying, that is, write a and b (in decimal, including any leading zeros necessary) and in each position add the numbers modulo 10, so for example 2456 ∗ 824 = 2270. (b) The set of functions f : R → R such that f (1) = 0 with pointwise addition (i. e. (f + g)(x) = f (x) + g(x)). (c) The set of real numbers with the operation x ∗ y = (^) xxy+y.
  2. How many generators are there in the cyclic group Z 28? [2 mins]
  1. Which of the following are subgroups of Z × Z? [15 mins]

(a) The set of all pairs (a, b) where a is divisible by 6. (b) The set of all pairs (a, b) such that a + 3b = 0. (c) The set of all pairs (a, b) such that 2a + b = 2. (d) The set of all pairs (a, b) such that 5a + 2b is divisible by 4. (e) The set of all pairs (a, b) such that a^2 + b^2 is a square number (i.e. a^2 + b^2 = c^2 for some c ∈ Z.) (f) The set of all pairs (a, b) such that a > b.

  1. Which of the following are subgroups of the group of permutations of the 6 element set { 1 , 2 , 3 , 4 , 5 , 6 }? [5 mins] (a) The set of permutations σ such that σ(1) + σ(4) + σ(5) = 10. (b) The set of permutations σ that either fix the set of odd numbers of send it to the set of even numbers. That is: either σ({ 1 , 3 , 5 }) = { 1 , 3 , 5 } or σ({ 1 , 3 , 5 }) = { 2 , 4 , 6 }.
  2. (a) Describe the subgroup of Z × Z 12 generated by (2, 8). [2 mins]

(b) Describe the subgroup of Z × Z 12 generated by (2, 8) and (3, 4). [ mins]

  1. (a) Write σ =

as a product of disjoint cycles. [2 mins] (b) What is the order of σ? [1 min] (c) Is σ an odd or even permutation? [2 mins] (d) Which of the following permutations are conjugate to σ in S 9? [3 mins]

(i)

(ii)

(iii)

  1. Draw the Cayley graph of A 4 with generators (123) and (234). [8 mins]
  2. Which of the following subgroups are normal? [14 mins]

(a) The subgroup of the group of symmetries of a hexagon generated by a 120◦^ rotation. (b) The subgroup of the group of symmetries of a hexagon generated by a 180◦^ rotation. (c) The subgroup of the additive group of real numbers generated by the numbers whose square is rational.

  1. Let H 6 G. Show that the commutator subgroup of H is a subgroup of the commutator subgroup of G, and that the centre Z(H) contains Z(G) ∩ H. [6 mins]