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Midterm Exam (Total 74 Points)

MTH 330 Abstract Algebra I

1. (6 Points, 2 each) Provide the definition of the followings.

1.1) A group

(G,*)

1.2) A group homomorphism from the group (G,*) to the group (H,®)

1.3) A cyclic group G

Name ... I.D ... .

2. (10 Points, 2.5 each) Provide a simple example of the group in each problem below. If there is no such example exist, provide your reason why it does not exist.

2.1) A non-cyclic group of order 4

2.2) A non-abelian group of order 24

2.3) A cyclic subgroup of order 8 of the cyclic group ( Z_{36 ,}

+)

:? .. .4) An abelian group of order 8 which is not cyclic

Name ... I.D ... .

3. (Total 10 Points) Let

c-{[~ ; ]

^{n E Z }·}

3.1) (6 Points) Assume that the matrix multiplication is associative. Show that G is a group under matrix multiplication.

Name ... I.D ... .

3.2) (4 Points) Show that G is isomorphic to

(Z,+).

• Name ... I.D ... .

4. (9 Points) If G is any group, define a function <p: G ~ G by q7(g)

=

g-^1• Show that G is abelian if and only if <p is a homomorphism.

..

Name ... I.D ... , ... .

5. (Total 8 Points) Let S, T be subgroups of a group G . 5. I) (5 Points) Show that S nT is a subgroup of G.

5.2) (3 Points) Is it also true that SuT is always a subgroup of G? (Provide a proof if it is true, or provide a counterexample if it is false.)

•

Name ... B • • • • • • • • • • , . . . 1.0 ... .

6. (Total 10 Points, S each) Let (O,•) and {H,ISJ>) be two groups, Let 0₀ be a subgroup of G and H_O be a subgroup of H. Suppose that ; : G -+ H be a group homomorphism.

6.1) Show that ¢(G_{0 )} := { ¢(x)

Ix

e G_{0 }} is a subgroup of H.

•

Name ... 1.0 ... .

6.2) Show that ¢-¹

(H

0) := { x E GI ¢(x) E H_{0 }} is a subgroup of G.

•

Name ... 1.0 ... .

7. (Total 10 Points, 5 each) Let G be a group with identity e and let g e G be any element of the group G.

7 .1) Show that if o(g) = k and m is another integer such that gm

=

e, then k

I

m.

7.2) Suppose further that G is cyclic of order n. Show that o(g)J n. ( Hint: To prove 7.2), you may refer to the result of 7.1) even if you are not able to

prove 7.1))

Name ... I.D ... ..

8. (11 Points) How many subgroups does ( Z ₅₄,

+)

have? List all subgroups of ( Z _{54 ,}

+)

and also provide the lattice diagram of subgroups of ( Z ₅₄,

+) .