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Material Type: Exam; Class: Group Theory; Subject: Mathematics; University: Northeastern University; Term: Fall 2010;
Typology: Exams
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Prof. Alexandru Suciu MATH 3175 Group Theory Fall 2010
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(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?
(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?
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?
, 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.
(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?
(b) A 4 and D 12.
(c) S 4 and D 6 × Z 2.
(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.
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 )?
(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).