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What does it mean?

Write down all possible cycle structures (all possible products of independent cycles) in

S_{5}

.

Does it mean write out all possible cycles: e, (12), (13), (14),...,(1524), (1425),...,(12)(34),...

Or does it mean: e, (xx), (xxx), (xxxx), (xxxxx), (xx)(xx), (xx)(xxx)?

:hmmmm2:

Reply 1

SimonM

The second one. Emphasis on structures.

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Reply 2

Mark85

Remember that every element of

S_n

can be written uniquely as a product of disjoint cycles (up to order obviously since they commute).

Therefore, the cycle structure of an element is essentially the lengths of the cycles in such a decomposition. You could therefore write it as a tuple* of numbers e.g.

(1 2 3) (4 5)

has cycle structure

(2,3)

since it is the product of a 2-cycle and a 3-cycle

*strictly speaking, if you do this, you have to fix an order e.g. I went from low to high so wrote (2,3) instead of (3,2) but this is being pedantic.

Reply 3

RamocitoMorales

Original post by Mark85

Remember that every element of

S_n

(1 2 3) (4 5)

has cycle structure

(2,3)

I wrote:

(1), (2), (3), (4), (5), (2,2), (2,3)

where

(i_{1},...,i_{n})

represents the composition of lengths of disjoint cycles,

i_{k}

.

Is that correct?

(edited 11 years ago)

Reply 4

RamocitoMorales

So basically, I've said that

S_{5}

has cycle structures (all possible products of independent cycles), which can be written as a composition of list of lengths,

(1),(2),(3),(4),(5),(2)(2),(2)(3)

Then it asks,

For each cycle structure, write one representative of the corresponding conjugacy class.

So for

(1)

I wrote

(e)

, for

(2)

I wrote

(12)

and so on....then for

(2)(2)

I wrote

(12)(45)

and for

(2)(3)

I wrote

(13)(24)

.

Then it asks,

For each cycle structure, give the size of the corresponding conjugacy class, i.e. the number of elements in each class (You need not show your work).

What is it asking here? Is the 'size' just the 'list of length', which I already worked out anyway? I'm rather boggled. :hmmmm2:

Reply 5

Mark85

Original post by RamocitoMorales

So basically, I've said that

S_{5}

has cycle structures (all possible products of independent cycles), which can be written as a composition of list of lengths,

(1),(2),(3),(4),(5),(2)(2),(2)(3)

This is correct although I would write

(2,2)

and

(2,3)

for the last two but as long as you carefully explain your notation (what you said in the previous post about this doesn't quite make sense to me), this doesn't matter.

Original post by RamocitoMorales

So for

(1)

I wrote

(e)

, for

(2)

I wrote

(12)

and so on....then for

(2)(2)

I wrote

(12)(45)

and for

(2)(3)

I wrote

(13)(24)

Ok.

Then it asks,

For each cycle structure, give the size of the corresponding conjugacy class, i.e. the number of elements in each class (You need not show your work).

What is it asking here? Is the 'size' just the 'list of length', which I already worked out anyway? I'm rather boggled. :hmmmm2:

Read it more carefully... especially the bit after i.e.

A conjugacy class is in particular a set of elements from your group. The size of a conjugacy class is the number of elements in it.

So, for example, the size of the conjugacy class of

(1 2)

is the number of elements in the same conjugacy class as

(1 2)

i.e. the number of elements conjugate to it.

(edited 11 years ago)

Reply 6

RamocitoMorales

Original post by Mark85

(1 2)

is the number of elements in the same conjugacy class as

(1 2)

i.e. the number of elements conjugate to it.

Ah, so for the representative of a conjugacy class corresponding to cycle structure

(2)

(12)

, the 'size' would be 10, as there are 10 elements in the same conjugacy class (i.e.

(13),(14),...,(23),...

), right?

So for cycle structure

(1)

I'd have size of conjugacy class 1, for

(2)

I'd have size of conjugacy class 10,...

(edited 11 years ago)

Reply 7

Mark85

Original post by RamocitoMorales

Ah, so for the representative of a conjugacy class corresponding to cycle structure

(2)

(12)

, the 'size' would be 10, as there are 10 elements in the same conjugacy class (i.e.

(13),(14),...,(23),...

), right?

Yeah. No need to put size in inverted commas though... when you are talking about a finite set - size always means the number of elements in it. This isn't a foreign concept or anything that is particular to conjugacy classes.

Reply 8

RamocitoMorales

Original post by Mark85

Yeah. No need to put size in inverted commas though... .

Force of habit. :colondollar:

Reply 9

RamocitoMorales

Original post by Mark85

Am I right in thinking that the size of the conjugacy class of cycles with the longest length

(5)

(n-1)!

And how would I find out the size of the rest?

Thank you.

Reply 10

Mark85

Original post by RamocitoMorales

Am I right in thinking that the size of the conjugacy class of cycles with the longest length

(5)

(n-1)!

Yes.

And how would I find out the size of the rest?

Thank you.

If you don't know a formula for the size of the conjugacy classes in

S_n

- use the same method that you used for the other two... just basic combinatorics.

Reply 11

RamocitoMorales

Original post by Mark85

If you don't know a formula for the size of the conjugacy classes in

S_n

- use the same method that you used for the other two... just basic combinatorics.

Okay, I worked out the sizes for all of the conjugacy classes, but now it's asking me:

For each cycle structure, say whether the elements of the correspinding class are even or odd, i.e. whether the elements are in

A_{5}

or not.

I've said:

Let

\sigma \in S_{5}

\pi(\sigma)=

# of entries of the list of

\sigma

The parity

p(\sigma)=5-\pi(\sigma)

.

Consider

p(\sigma) \mod{2}

Call

•

$\sigma$ even if $p(\sigma)\mod{2}\equiv 0$

•

$\sigma$ odd if $p(\sigma)\mod{2}\equiv 1$

...

This might seem a 'stupid' question, but how do I find

\pi(\sigma)

for each of the conjugacy classes? :colondollar:

Reply 12

Mark85

Original post by RamocitoMorales

I've said:

Let

\sigma \in S_{5}

\pi(\sigma)=

# of entries of the list of

\sigma

What does 'number of entries of the list of

\sigma

' even mean?

Why are you complicating this anyway?

You wrote down a representative for each conjugacy class - there are only a few... it is then straightforward to check directly whether they are odd or even.