Page 55 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes Cyclic Groups: Let G be any group and S a subset of G. Consider the family F of all subgroups of
G that contain S, that is, F = { H | H  G and S  H }.

3.6 Review Questions

1. Find all cyclic subgroups of Z .
×
24
2. In Z , find two subgroups of order 4, one that is cyclic and one that is not cyclic.
×
20
(a) Find the cyclic subgroup of S generated by the element (1, 2, 3)(5, 7). (b) Find a
7
subgroup of S that contains 12 elements. You do not have to list all of the elements if you
7
can explain why there must be 12, and why they must form a subgroup.
3. In G = Z , show that
×
21
4. H = { [x] | x  1 (mod 3) } and K = { [x] | x 1 (mod 7) } are subgroups of G.
21
21
5. Let G be an abelian group, and let n be a fixed positive integer. Show that
N = { g in G | g = a for some a in G } is a subgroup of G.
n
6. Suppose that p is a prime number of the form p = 2 + 1.
n
(a) Show that in Z the order of [2] is 2n.
×
p
p
(b) Use part (a) to prove that n must be a power of 2.
7. In the multiplicative group C of complex numbers, find the order of the elements
×
2 2 2 2
 =   i and  =   i .
2 2 2 2
8. Let K be the following subset of GL (R).
2


   a b 
0

K =  c d  d  a,c   2b,ad bc  
     
9. Show that K is a subgroup of GL (R).
2
10. Compute the centralizer in GL (R) of the matrix   1 0  .
2
 1 0 

   m b 

0 .
11. Let G be the subgroup of GL (R) defined by G =  0 1  m  
2
     
12. Let A =   1 1  and B =    1 0  . Find the centralizers C(A) and C(B), and show that C(A)
 0 1   0 1 
C(B) = Z(G), where Z(G) is the center of G.

Answers: Self Assessment

1. (b) 2. (a) 3. (a) 4. (d) 5. (a)

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