Assignment:

Hand-in-homework: Find the subgroup generated.

(1)  < (34) >  = {I,(34)}         

The subgroup generated by a 2-cycle has 2 elements. {I,(34)}.

(2)  < (1234) > = {I,(1234),(13)(24),(1432)}     

The subgroup generated by a 4-cycle has four elements.

(3)  < (12),  (34) >     = {I,(12),(34),(12)(34)}

Since disjoint cycles commute, and (12) and(34) both

square to I, it is easy to see that this set is closed.

(4)  < (1234),  (13) >  = Symmetries of the square.  

Since the symmetries of the square contain both of these

permutations, one can not get anything larger than a

subgroup with eight elements.  The four cycle alone

generates 4 of them, and multiplying each of them by

(13) generates 4 more.

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


(5)  < (123),   (234) >   = A4

Since three cycles are even, we cannot hope to get anything

Larger than A4.  We know we have 5 elements already.

I,(123),(132),(234),(243).  If we multiply these together we

get more elements:

(123)(234) = (12)(34)
(123)(243) = (124)
(132)(234) = (134)
(132)(243) = (13)(24) 

Since we have 2 of the 2-2-cycles, we get all three 2-2-cycles.
Since we have two more 3-cycles, we get them and their inverses
giving us 7 more elements.  This is what we need to get all of A4.


We can also argue that if we can cyclically shift the first three

and the last three, we can put any choice we want into positions

3 and 4.  This gives us at least 12 permutations which is 

enough to show that the two three cycles generate A4. 

(6)  < (12)(34),  (123) >  = A4

Since both permutations are even, we can not get anything larger

than A4.  Since

(12)(34)(123) = (1)(243)  we have (123) and (243) which means

we have (123) and (234) which by part (5) generate A4.


(7)  < (234), (34) >    = symmetries of a triangle.

Both of these leave position 1 fixed. Thus you can never

get more than the six permutations of "2","3","4".

(234)(234) = (243)
(234)(34)  = (23)
(243)(34)  = (24)

Here are the other 3 permutations of the symmetries of a triangle. 


(8)  < (123), (132) >     =  {I,(123,(132}

This is the subgroup generated by a 3-cycle