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Final Exam Fall 2010 - Group Theory | MATH 3175, Exams of Mathematics

Material Type: Exam; Class: Group Theory; Subject: Mathematics; University: Northeastern University; Term: Fall 2010;

Typology: Exams

2010/2011

Uploaded on 06/02/2011

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Name:
1 2 3 4 5 6 7 8 9 10 11 12 Σ
Prof. Alexandru Suciu
MATH 3175 Group Theory Fall 2010
Final Exam
1. Let Gbe the group defined by the following Cayley table.
12345678
1 12345678
2 25476183
3 38527416
4 43658721
5 56781234
6 61832547
7 74163852
8 87214365
(a) For each element aG, find: the order |a|; the inverse a1; and the centralizer C(a).
a12345678
|a|
a1
C(a)
(b) What is the center of G?
pf3
pf4
pf5
pf8

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Name:

1 2 3 4 5 6 7 8 9 10 11 12 Σ

Prof. Alexandru Suciu MATH 3175 Group Theory Fall 2010

Final Exam

  1. Let G be the group defined by the following Cayley table.

1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 7 8 2 2 5 4 7 6 1 8 3 3 3 8 5 2 7 4 1 6 4 4 3 6 5 8 7 2 1 5 5 6 7 8 1 2 3 4 6 6 1 8 3 2 5 4 7 7 7 4 1 6 3 8 5 2 8 8 7 2 1 4 3 6 5

(a) For each element a ∈ G, find: the order |a|; the inverse a−^1 ; and the centralizer C(a).

a 1 2 3 4 5 6 7 8

|a|

a−^1

C(a)

(b) What is the center of G?

  1. Let G be an abelian group with identity e, and let H be the set of all elements x ∈ G that satisfy the equation x^2 = e. Prove that H is a subgroup of G.
  2. Let G = 〈a〉 be a group generated by an element a of order |a| = 30. (a) Find all elements of G which generate G.

(b) List all the elements in the subgroup 〈a^6 〉, together with their respective orders.

(c) What are the generators of the subgroup 〈a^6 〉?

(d) Find an element in G that has order 3. Does this element generate G?

  1. Let α =

[

]

and β =

[

]

, viewed as elements in the symmetric group S 7. (a) Compute the products

βα =

αβ =

(b) Compute the inverses

α−^1 =

β−^1 =

(c) Compute the conjugate of β by α:

αβα−^1 =

(d) Do α and β commute?

  1. Let α =

[

]

, viewed as an element in S 9.

(a) Write α as products of disjoint cycles.

(b) Find the order of α.

(c) Write α as a product of transpositions.

(d) Find the parity of α.

  1. Let R be the additive group of real numbers, and let R∗^ be the multiplicative group of non-zero real numbers. Consider the map φ : R → R∗^ given by φ(x) = ex. (a) Show that φ is an homomorphism from R to R∗.

(b) What is the kernel of φ?

(c) What is the image of φ? For each y ∈ im(φ) find an x ∈ R such that φ(x) = y?

(d) Is φ injective (i.e., one-to-one)?

(e) Is φ surjective (i.e., onto)?

(f) Is φ an isomorphism?

  1. Show that the following pairs of groups are not isomorphic. In each case, explain why. (a) U (15) and Z 8.

(b) A 4 and D 12.

(c) S 4 and D 6 × Z 2.

  1. (a) List all abelian groups (up to isomorphism) of order 100. Write each such group as a direct product of cyclic groups of prime power order.

(b) Let G be an abelian group of order 100. Suppose that G has exactly 3 elements of order 2, and 4 element of order 5. Determine the isomorphism class of G.

  1. Let H be set of all 2 × 2 matrices of the form

[

a b 0 d

]

, with a, b, d ∈ Z 3 and ad 6 = 0. (a) Show that H is a subgroup of GL 2 (Z 3 ).

(b) Is H a normal subgroup of GL 2 (Z 3 )?

  1. Let α : G → H and β : H → K be two homomorphisms. (a) Show that β ◦ α : G → K is a homomorphism.

(b) Show that ker(α) is a normal subgroup of ker(β ◦ α).

(c) Show that im(β ◦ α) is a subgroup of im(β).