Math Is Fun Forum

cadjeff

Member

Registered: 2007-05-08

Posts: 26

The group S4 - group theory

hello,

even after attending all my lessons in this subject, i'm still useless at it.
Some help on the following would be much appreciated.

1. Let F be a non square rectangle with the four corners labelled 1-4.
Identify the subgroup H1 of S4 which corresponds to S(F).

2.By finding appropriate plane figures, show that each of the following is a subgroup of S4.
a) H3=(  e, (1234), (13)(24), (1432)   )
b) H4=(  e, (123), (132), (12), (13), (23)   )

3. What are possible orders of subgroups of S4? ...explain.

4. Says to list each possible cycle structure of an element of S4. For each cycle structure, determine how many elements of S4 have the structure and state the order of each such element.

5. Write down all the cyclic subgroups of S4.

6. List all the subgroups of S4 which are isomorphic to K4.
Explain why your list is complete; that is, explain why there are no other subgroups of S4  isomorphic to K4.

7. Explain why S4 does not have cyclic subgroup of order 6.
Find all four subgroups of S4 of order 6.
Hint: Question 2 may help here.
You may assume that S4 has four subgroups of order 6.

There's more but i think this is more than enough to put up for now.

Any pokes at some of these questions would be a great help.

Thanks for your time,

caddy

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: The group S4 - group theory

1. What does the notation S(F) mean?

2. a is a rectangle, b is a tetrahedron.  It should be easy to figure out what exactly you're doing to each from there.  Just draw each shape and label the vertices.  Now draw a new shape and perform the permutation on them.

3. Lagrange's theorem

4. Cycle structure just means the number of k-cycles.  So for example, (12)(346)(57) has 2 2-cycles, and 1 3-cycle.  Remember that an element has order k if and only if the lcd of it's cycle lengths is k.

5. Just simply pick a single element, and generate all the possible elements.  There are only 24 elements in S4, so this isn't too bad.  You should start to find a patter in it which will speed up some computations.

6. K4?  Klein group of order 4?  You need subgroups of order 4.  What does this mean about the cycle structure of each element in the group? (Note, that an elements order must divide the order of the subgroup it lies in)

7, Cycle structure!  How many elements does it take to get cycles whose lcd is 6?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."