math test

MIDTERM

e stands for identity element.

(1) State the structure theorem for finite generated Abelian group. Using this theorem to classify all finite generated Abelian group of order 120.

(2) Prove that Zm ×Zn is a cyclic group if and only if gcd(m,n) = 1. Find all generators of cyclic group Z5 ×Z4.

(3) What is the order of (3, 2) in group Z12 ×Z6 and prove it. (4) Prove that Z×Z/〈(2, 2)〉 is an infinite group but is not an infinite cyclic group. (5) Make a list of all homomorphisms Z9 → Z9. Prove that this is a complete list. Is there

a homomorphism from Z8 → Z5? (If yes, list all possible homomorphism) Is there a homomorphism from Z10 → Z5 (If yes, list all possible homomorphism).

(6) Find the order of the following permutations: (a) (145)(2345) ∈ S5. (b) (154)(254)(1234) ∈ S5. (c) (1574)(324)(3256) ∈ S7

(7) Define a map φ1 from Z2 ×Z → Z4 as follows: φ1(m,n) = 2mn. Define a map φ2 from Z10 ×Z9 → Z6 as follows: φ2(m,n) = mn. Is φ1,(φ2) a well-defined map? Is it a group homomorphism? If it is a homomorphism, what is ker(φ)

(8) If H1,H2 are 2 subgroups of G, prove that H1 ∩H2 is also a subgroup of G. If further assume both H1,H2 are normal subgroups of G, is H1 ∩H2 a normal subgroup of G. If yes, prove it. If no, give a counterexample.

Date: May 28, 2021.

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