Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
Material Type: Exam; Class: Group Theory; Subject: Mathematics; University: Northeastern University; Term: Fall 2010;
Typology: Exams
1 / 8
This page cannot be seen from the preview
Don't miss anything!
Name:
1 2 3 4 5 6 7 8 9 10 11 12 Σ
Prof. Alexandru Suciu MATH 3175 Group Theory Fall 2010
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?
(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).
