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MATH 3360 Exam I - Problems and Solutions, Exams of Algebra

Problems and solutions from exam i of math 3360. Topics covered include group theory, binary operations, and subgroups. Students are required to show their work for each problem.

Typology: Exams

Pre 2010

Uploaded on 03/10/2009

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MATH 3360 Exam I February 15, 1995
Show your work for each problem. You do not need to rewrite the statements of the
problems on your answer sheets.
1. Assume that a : S 6 T and ß : T 6 U. Consider the following statement:
If ß is not onto, then ßBa is not invertible. (1)
(a) Is statement (1) true? Why or why not?
(b) State the converse of statement (1).
(c) Is the converse of statement (1) true? Why or why not.
2. Let E be the set of even integers. Let .
(a) Show that * is an operation on E.
(b) Is * associative? Why or why not?
(c) Show that * has an identity.
3. Is (G,*) a group? Why or why not?
(a) G = { e * n 0 Z }; * is multiplication.
n
(b) G = Z c { z * 1/z 0 Z } c { 0 }; * is addition.
4. Write (2 3 5)(1 2 5 3)(1 3 4) as a single cycle or a product of pairwise disjoint
cycles.
5. Complete: If G with operation * is a group, then G is non-Abelian iff a*b b*a .
. . .
6. Show that S is non-Abelian.
4
7. It can be shown that ú with operation + being component-wise addition, i.e., (a,b)
2
+ (c,d) = (a+c,b+d), is a group, . Let L = { (x,y) * 2x - 3y = 0 }, i.e.,
L is the set of points which lie on the line satisfying the equation 2x - 3y = 0.
Show that (L,+) is a subgroup of (ú,+).
2
8. Identify the symmetry group for the following figure,
which consists of a square with one diagonal and the
middle third of the other diagonal.

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MATH 3360 Exam I February 15, 1995 Show your work for each problem. You do not need to rewrite the statements of theproblems on your answer sheets.

  1. Assume that a : S 6 T and ß : T 6 U. Consider the following statement: If ß is not onto, then ßBa is not invertible. (1) (a) Is statement (1) true? Why or why not?(b) State the converse of statement (1). (c) Is the converse of statement (1) true? Why or why not.
  2. Let E be the set of even integers. Let. (a) Show that * is an operation on(b) Is * associative? Why or why not? E. (c) Show that * has an identity.
  3. Is ( G ,*) a group? Why or why not? (a) G = { e n * n 0 Z }; * is multiplication. (b) G = Z c { z * 1/ z 0 Z } c { 0 }; * is addition.
  4. Write (2 3 5)(1 2 5 3)(1 3 4) as a single cycle or a product of pairwise disjointcycles.
  5. Complete: If G with operation * is a group, then G is non-Abelian iff a*b Ö b*a. ...
  6. Show that S (^) 4 is non-Abelian.
  7. It can be shown that ˙ 2 with operation + being component-wise addition, i.e., ( a,b )
    • ( L is the set of points which lie on the line satisfying the equation 2 c,d ) = ( a+c,b+d ), is a group,. Let L = { ( x,y ) * 2 x - 3 y = 0 }, i.e., x - 3 y = 0. Show that ( L ,+) is a subgroup of (˙ 2 ,+).
  8. Identify the symmetry group for the following figure, which consists of a square with one diagonal and themiddle third of the other diagonal.