Math 322, Fall Term 2011
Final Exam
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Solutions to Problem set 4
Problem 1.7.3. Need to check the axions of a group action:
(1) (r + s) (x, y) = (x + (r + s)y, y) = r (x + sy, y) = r (s (x, y),
and
(2) 0 (x, y) = (x + 0y, y) = (x, y) for every (x, y) R2 .
Both axioms check out, so this is a g
Solutions to Problem set 3
Problem 3.1.5. The order of gN in G/N is the smallest positive
integers n such that (gN )n = N in G/N or equivalently, g n N = N or
equivalently, g n N .
To construct an example, where the order of gN in G/N is strictly
smaller
Solutions to Problem set 1
Problem 1.6.8. These groups have dierent orders, | Sm | = m! and
| Sn | = n!.
Problem 1.6.9. D24 has an element of order 12, so it suces to
show that S4 does not have an element of order 12.
Indeed, the possible decompositions o
Solutions to Problem set 1
Problem 1.11. The order of x Z/12Z is the smallest positive
integer n such that nx is divisible by 12 or, equivalently, such that nx
is divisible by both 3 and 4. Write x = 2a 3b n0 , where a, b 0 and n0
is not divisible by eith
The University of British Columbia
Final Examination - December 13, 2010
Mathematics 322
Closed book examination
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Time: 2.5 hours
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THE UNIVERSITY OF BRITISH COLUMBIA
SESSIONAL EXAMINATIONS DECEMBER 2009
MATHEMATICS 322
Time: 2 hours 30 minutes
1. [16 points] Determine whether the following statements are true or false
(you have to include proofs/counterexamples):
(a) The rings Z/35Z
The University of British Columbia. Mathematics 322
Final Examination - Monday, December 10, 2012, 3:30-6pm. Instructor: Reichstein
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Every problem is worth 5 points.
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THE UNIVERSITY OF BRITISH COLUMBIA
SESSIONAL EXAMINATIONS DECEMBER 2008
MATHEMATICS 322
TIME: 2 1/2 hours
1. [16 marks]
a) Explain what is meant by the centre Z(G) of a group G.
b) Prove that Z(G) is a subgroup of G.
c) If a subgroup H of a group G is con
Solutions to Problem set 7
Problem 5.1.4. Suppose |A| = pa mA and |B| = pb mB , where p
does not divide mA or mB . Then |A B| = |A| |B| = pa+b mA mB .
Thus Sylow p-subgroups in A, B and A B have orders pa , pb and
pa+b , respectively.
Choose Sylow p-subgr