Math 451
Characters
Notation: Let G be a nite group of order N whose conjugacy classes are C1 , C2 , . . . Cr
where C1 = cfw_1. We know that G has exactly r inequivalent irreducible representations
call them i : G GL(Vi ) for i = 1, . . . , r and let 1 b
MATH 451 FIRST MID-TERM
NAME: John Q. Public
Question Marks
1
12
2
25
3
28
4
25
5
10
1
2
MATH 451 FIRST MID-TERM
Question 1. Let H be a nonempty subset of the group G. Prove that H is a
subgroup of G iff a b1 H for all a, b H .
First suppose that H is a s
MATH 451 SECOND MID-TERM
NAME: John Q. Public
Question Marks
1
20
2
20
3
20
4
20
5
20
1
2
MATH 451 SECOND MID-TERM
Question 1. Throughout this question, let p be a prime.
(a) Suppose that G is a nite p-group and that X is a nonempty G-set such
that |X | 0
MATH 451 FINAL EXAM
Question 1. Let G be a group and for each g G, let ig : G G be the map
dened by ig (x) = gxg 1 .
(a) Prove that ig Aut(G) for each g G.
(b) Prove that Inn(G) = cfw_ ig | g G is a subgroup of Aut(G).
(c) Prove that if Aut(G) and g G, t
Name
Math 451, 01, Exam #1 Oct., 2007
Be sure to show all your work. Unsupported answers will receive no credit.
Use the backs of the exam pages for scratchwork or for continuation of your
answers, if necessary.
Unless otherwise indicated, G is a group wi
Math 451, 01, Exam #1 October 2007
Answer Key
1. (20 points): If the statement is always true, circle True and prove it. If the statement
is never true, circle False and prove that it can never be true. If the statement is true in
some cases and false in
Name
Math 451, 01, Exam #2 Nov., 2007
Be sure to show all your work. Unsupported answers will receive no credit.
Use the backs of the exam pages for scratchwork or for continuation of your
answers, if necessary.
Unless otherwise indicated, G is a group wi
Math 451, 01, Exam #2
Answer Key
1. (25 points): If the statement is always true, circle True and prove it. If the statement
is never true, circle False and prove that it can never be true. If the statement is true in
some cases and false in others, circl
Math 451
A Note on Finite Fields
Definition: Let F be a (non-empty) set with two operations (addition and multiplication). Suppose that: F is an abelian group under addition. For all a, b, c F we have that a(b + c) = ab + ac and (a + b)c = ac + bc. F is a