Question

Help please! Write explicitly the subgroup of G generated by g:

G= S6 and g=(123)(46)

Answer 1

Well, figure out what is g^2, and g^3, g^4, g^5. We know that g^6 = 1, the identity, so there's only a few elements here

Answer 2

e.g.,
g^2 = (123)(46)(123)(46) = (123)(123)(46)(46)

Now (46)^2 = 1 (because 4 --> 6 --> 4 and 6 --> 4 --> 6)
What then is (123)^2 ?

Answer 3

is it (132)?

Answer 4

yes, exactly. So g^2 = (132). Now what is g^3?

Answer 5

so is the subgroup just {g, g^2...g^6}? whatever the values are?

Answer 6

(For the record, I'm going to change notation. Using 1 for the identity here is confusing; so let's use the standard e instead.)

Yes...

Answer 7

Given a group G and a member of that group g, then the subgroup generated by g is

{ g^n | n in Z }

In this case, we know that g^6 = e and therefore this set is at most six members
{e, g, g^2, g^3, g^4, g^5}

Answer 8

because g^n = g^(6+n) for every integer n, whether positive or negative.

Answer 9

and we know that because we are in S6

Answer 10

Hence it's sufficient to calculate the values of g, g^2, g^3, g^4, g^5 and see whether or not they are unique.

Answer 11

So that's why we're stepping through these calculations now to see what these group members are

Answer 12

ok. why is it important that they're unique?

Answer 13

Only because when we come to write down the set which represents the sub-group, we don't list any element twice.

Answer 14

For example, consider the sub-group generated by h = (123). Well h^2 = (132) and h^3 = e.

Answer 15

We can keep on calculating if we want to get h^4 and h^5. But as h^3 = e, it follows that h^4 = h and h^5 = h^2 and there we don't need to do the calculations.

Answer 16

Therefore the subgroup generated by h = (123) is
<(123)> = { e, (123), (132) }

Answer 17

Great. Thank you!

Answer 18

Correction/Clarification: it's not true that for any g in S6 that g^6 = e. Although it is true for your particular g.

The order of the group Sn -- the permutations of {1, 2, ..., n} -- is n!