# Group Theory Q

1. Feb 16, 2006

### ElDavidas

I was in a tutorial today and was asked

"What is the largest order that an element of $$S_{10}$$ can have?"

I thought the answer was 10! but I've been told this is wrong. Can someone help me out with what's going on? I thought you calulated the order by the formula:

$$|S_n| = n!$$

Last edited: Feb 16, 2006
2. Feb 16, 2006

### Muzza

You were asked about the order of an element, not the order of the group.

There can't be an element of order 10! in S_10, because then S_10 would be abelian (even cyclic).

Do you know that any permutation can be written as the product of disjoint cycles?

3. Feb 16, 2006

### JasonRox

I noticed that a lot of people can't answer these questions when asked.

The question that Muzza just asked is something you should know to answer the question you want to know.

4. Feb 16, 2006

### matt grime

Ahem, this seems that it is also a matter of English and presumption.

For the following question:

Let G be a group of order m, what is the largest order an element can have?

Then the correct answer really is m, since all elements have order dividing m and there is always a cyclic group of order m.

However, just because something can happen doesn't mean it does happen. If we're given the extra information that G is actually S_n and n!=m, then, we can get a *better* answer, and indeed we can explicitly say what all permissible orders of elements are.

Can is a bad word, in this question, or many questions. The better phrase would be: what is the largest order of an element of S_n.