Math 3721
Abstract Algebra I: Gift 1
Name:
Spring 2006
Score:
/105
1. (15 points) Let
= (12)(58)(346)(52)(41)(37)(67)
(a) Write as a product of disjoint cycles.
(b) Find |.
(c) Is A8 ?
(d) What is 1 ?
(e) What is the maximum possible order of an element
Math 3721 Code
Abstract Algebra I: Gift 2
Name:
Spring 2006
Score:
/100
1. (20 points) Let G = Z20 , and H =< 4 > .
(a) List the distinct left cosets of H in G.
(b) What is the order of 6 + H in G/H ?
(c) Is G/H V or is G/H Z4 ? Explain your answer.
=
=
2
Math 3721
Abstract Algebra I: Gift 3
Name:
Spring 2006
Score:
/100
1. (10 points)
(a) Give the denition of a eld.
(b) Give the denition of a zero divisor.
2. (15 points) Let G be a group. Show that Inn(G) is a normal subgroup of Aut(G).
3. (10 points)
(a)
Math 3721 Code 2236
Abstract Algebra I: Gift 1
Name:
Spring 2002
Score:
/100
1. (10 points) Give the denition of a group.
2. (25 points)
(a) Let G be a group. Give the denition of a subgroup H of G.
(b) Prove that
10
:cZ
H=
c1
is a subgroup of G = SL(2,
Math 3721 Code 2236 Name:
Abstract Algebra I: Gift 2 Score:
Spring 2002 /100
1. (20 points) Let G = Z8 , and H =< 4 > . (a) List the distinct left cosets of H in G. (b) What is the order of 6 + H in G/H ? (c) Is G/H = V or is G/H Z4 ? Explain your answer.
Math 3721 Code 2236 Name:
Abstract Algebra I: Gift 3 Score:
Spring 2002 /100
1. (10 points) Find the order of the indicated element in the indicated group: (a) (, 5) in S3 U (12). (b) (2, 4, 7) in Z8 Z5 Z10 . 2. (15 points) (a) Is the group Z2 Z24 cyclic?