Does A3 Demonstrate Both Commutative and Cyclic Properties?

[tgt]

Homework Statement

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).

Have I misread the question?

 

[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?

 

[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)}.

 

[Dick]

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

 

[tgt]

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?

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.

 

[tgt]

That subgroup has order 2, doesn't it?

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.

 

[Dick]

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

 

[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! that's question 13g. This is an excellent book for finding mistakes ... if you can find them.

 

[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.

 

[brydustin]

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.

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