Page 46 - DMTH403_ABSTRACT_ALGEBRA

Unit 3: Subgroups

Notes
Example: Consider G = M (C), the set of all 2 × 3 matrices over C. Check that (G,+) is an
2×3
abelian group. Show that

  0 a b  

S   0 0 c a, b, c C  is a subgroup of G.
   
Solution: We define addition on G By



a  b c  p q r   a p b q c r 
 d e f    s t u    d s e t u 
       f  
You can see that + is a binary operation on G. 0 =   0 0 0  is the additive identity and    a  c 
 0 0 0    d  f 
a  b c
is the inverse of    G.
 d e f 

Since, a + b = b + a  a, b  C, + is also abelian.
Therefore, (G,+) is an abelian group.
Now, since O  S, S  . Also for

 0 a b  0 d e
,
 0 0 c   0 0 f   S , we see that
   
 0 a b  0 d e  0 a d b e  
,
 0 0 c   0 0 f    0 0 c f   S.
      
= S I G.

Example: Consider the set of all invertible 3 × 3 matrices over R, GL (R). That is,
3
A  GL (R) iff det (A)  0. Show that SL (R) = (A GL (R) | det(A) = I ) is a subgroup of (GL (R),.).
3
3
3
3
Solution: The 3 × 3 identity matrix is in SL (R). Therefore, SL (R)  .
3
3
Now, for A, B  SL (R),
3
1
det (AB ) = det (A) det(B ) =  1, since det (A) = 1 and det (B) = I.
-1
-1
det(B)
-1
 AB  SL (R)
3
 SL (R) I GL (R).
3
3
Example: Any non-trivial subgroup of (Z, +) is of the form mZ, where m  N and
mZ = { mt | t  Z) = { 0,  m, ± 2m, ± 3m,....... ).
Solution: We will first show that mZ is a subgroup of Z. Then we will show that if H is a
subgroup of Z, H # {0}, then H = mZ, for some m  N.

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