301 Abstract Algebra
2:00 - 2:50 MWF
Hentzel 432 Carver
hentzel@iastate.edu
Web Page
http://www.math.iastate.edu/hentzel/class.301

Text: A First Course in Abstract Algebra, sixth Edition 

by John B. Fraleigh

Course: Chapter 1,2,3,4,5 

The material we are covering today is on Page 65-70 

Section 1.4 beginning with "Subgroups".   


Main Idea: Groups have other smaller groups inside them. 

Key Words: Subgroup, Kleinfour group, K4, A4, S4

Proper subgroup, trivial subgroup, lattice, Cyclic,  

Goal: Learn about subgroups. 

Definition of Subgroup:
----------------------

Let G be a group, S a subset of G.  Then S is a subgroup 

of G if under the operations of G, S itself is a group.


Lemma: A nonempty subset S of a group G is a subgroup of

G if and only if:
 
(a)  S is closed

(b)  S has inverses.


Proof: ===>  If S is a subgroup then the product of two 

elements of S is again in S.  This is called closure.  S 

also has inverses.  Therefore S does satisfy both (a) and (b).

      <===  Suppose S satisfies (a) and (b).  From closure, 

S inherits the associativity of the product in G.  Since S 

is not empty, there is some element s in S.  Therefore s' is

in S by (b) and s s' = e is in S from (a).  Therefore S has 

the identity.  From (b) directly we have that S has inverses.  

Since S has a product which is associative, has an identity, 

and inverses, S is a group.








S4 = {the set of permutations on four objects}
   
        1   I
        6   ( , )
        8   ( , , )
        6   ( , , , )
        3   ( , )( , )
    -------
       24 = 4! = number of permutations on 4 objects. 

A4 = {the set of even permutations on four objects}
        
        1    I
        8    ( , , )
        3    ( , )( , )
      ------
       12  which is half the elements in the group.   

K4 =  {I and set of 2-2-cycles}

        1    I  
        3  ( , )( , )
      ----
       4 elements  
 
The symmetries of the square.  There are three of these in S4,

they consists of a 4 cycle and its inverse, the 2-2-cycles, and

two specially selected 2 cycles.  For example:

   { I, (1234), (1432), (12)(34), (13)(24),(14)(23), (13), (24) } 


The symmetries of a triangle.  These are are obtained by choosing

one object and considering all the permutations of the other three.

For Example:

   { I, (123), (132), (12), (13), (23) }.  

Everything which leaves "4" fixed.


The subgroups generated by a single permutation.

      {I, (12)}
      {I, (123), (132) }
      {I, (1234), (1324), (1432)}
      {I, (12)(34) } 

Now the simplest subgroup to examine is the subgroup generated

by a single element.  It consists of a, a^2, a^3, a^4, ... and

so on.  If you are talking about the integers, you would get

1,2,3,4, ... and proceeding in this way you would not get a

subgroup because you would never get the inverses.  But in the

finite situation, you have to go in a circle.  Eventually you

will hit some power of a which is e, and then the next power

of a is a.   That is, with a^n = e, then a^(n+1) = a and you

start the circle all over again. 

                
        .    .   .    a^4

      .                    a^3
            
       .                   a^2
           a^n=e     a


Proof:  If there are only a finite number of elements in the

group, there must be some double counting.  That is,


a^n = a^m  when n > m.  Now multiply by a' m times giving

a^(n-m) = e.  Thus some power of a is e.  Define the

order of a, o(a) to be the smallest power n such that a^n = e.


(a)  What is the order of a 2-cycle 
                            3-cycle
                            4-cycle
                            5-cycle
                            n-cycle
                         ( , )( , ,) cycle

(b)  What is the order of a perfect shuffle?  How many times do

you have to shuffle the cards to return them to their original

positions.

(c)  The order of 1 in the integers is infinity.  o(1) = Infinity.


(d)  In Z12 what is the subgroup generated by 2? 

                    the subgroup generated by 3?  

                    the subgroup generated by 5? 

                    the subgroup generated by 7? 


Subgroups generated by more than one element.

(a) What is the subgroup generated by { (12), (1234) }?

(b) What is the subgroup generated by { (13), (1234) }?

(c) What is the subgroup generated by { (123), (234) }?

(d) What is the subgroup generated by { (14),  (23)  }?















Assignment:

Hand-in-homework: Find the subgroup generated.

(1)  < (34) >           

(2)  < (1234) >     

(3)  < (12),  (34) >    

(4)  < (1234),  (13) >   

(5)  < (123),   (234) >  

(6)  < (12)(34),  (123) > 

(7)  < (234), (34) >   

(8)  < (123), (132) >