Group theory: True or false

1. Jan 26, 2009

tgt

1. The problem statement, all variables and given/known data
Every nontrivial subgroup H of the symmetric group with 9 elements containing some odd permutation contains a transposition.

It does seem the case that if a subgroup of H of the symmetric group with 9 elements contain an odd permutation then certainly a transposition must be apparent (there might be more but surely one is apparent).

2. Jan 26, 2009

Dick

There are odd elements that don't consist of a single transposition. Some are the product of three transpositions. Or more. And there is no symmetric group with 9 elements. So you must mean H has 9 elements. And if H contains a transposition then it has a element of order 2. Or do you mean S_9? Still not true. The more I think about this the less sense it makes. Are you sure that's the real question?

Last edited: Jan 26, 2009
3. Jan 27, 2009

tgt

Maybe I have. It's asking whether every nontrivial subgroup of S9 containing an odd permutation must contain a single transposition. The answer is no if we consider the group {I, (12)(34)(56)}.

4. Jan 27, 2009

Dick

Ok, right. That's a subgroup of S9 and contains no transposition. I'm still fixated on H being having 9 elements. Sorry.

Last edited: Jan 27, 2009
5. Jan 27, 2009

tgt

symmetric group with 9 element is S_9 which certainly exits. Why do you say it doesn't?

http://en.wikipedia.org/wiki/Symmetric_group

The answer to the OP is false as shown by a counter example above.

6. Jan 27, 2009

tgt

Yes.

You've probably mistaken the wording in the OP. The symmetric group with 9 elements is obviously not good use of words. I really mean S_9 which has 9! elements.

7. Jan 27, 2009

Dick

I agree. There is a subgroup of order 2 with no transposition. As you're example points out. I edited the previous reply.

8. Apr 5, 2011

brydustin

This is an interesting problem:
Its Question 13d of "A First Course in Abstract Algebra by John B Fraleigh".
Of course the solution in the back of the book is wrong because it says that the statement is false; when in fact it is true. He also says that A_3 is a commutative group WHEN ITS CLEARLY NOT! thats question 13g. This is an excellent book for finding mistakes ..... if you can find them.

9. Apr 6, 2011

Deveno

A3 IS commutative, as is any group of order 3.

A3 = {I, (1 2 3), (1 3 2)}

it is cyclic, since 3 is prime.

10. Apr 6, 2011

brydustin

Yeah you're right .... thanks for pointing that out. My bad