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## Homework Statement

Let [itex]G=S_6[/itex] acting in the natural way on the set [itex]X = \{1,2,3,4,5,6\}[/itex].

(a)(i) By fixing 2 points in [itex]X[/itex], or otherwise, identify a copy of [itex]S_4[/itex] inside [itex]G[/itex].

(ii) Using the fact that [itex]S_4[/itex] contains a subgroup of order 8, find a subgroup of order 16 in [itex]G[/itex].

(b) Find a copy of [itex]S_3 \times S_3[/itex] inside [itex]S_6[/itex]. Hence, or otherwise, show that [itex]G[/itex] contains a subgroup of order 9.

(c) Find a subgroup of order 5 in [itex]G[/itex].

## The Attempt at a Solution

How do I find a subgroup of S6 which is isomorphic to S4 by fixing 2 elements of X?

For (a)(i) I know:

[itex]S_4 = \{ e, (12), (13), (14), (23), (24), (34), (123), (132), (142),[/itex]

[itex]\;\;\;\;\;\;\;\;\;\;\;\;(124), (134), (143), (234), (243), (1234), (1243), (1324),[/itex]

[itex]\;\;\;\;\;\;\;\;\;\;\;\;(1342), (1423), (1432) , (12)(34) , (13)(24) , (14)(23) \}[/itex]

As written, this can be regarded as a subgroup of S6.

Now, if I want to fix 5 and 6, what permutations in S6 does that leave me with? And conversely, if I write S4 as I've done above, what is the action of each element of S4 on 5 and 6?

For (a)(ii) I know that the following is a subgroup of [itex]S_4[/itex] of order 8: [tex] \{ e, (24), (13), (12)(34), (14)(23), (1432), (1234), (1324) \}[/tex]

How do I find the subgroup of order 16 by building it up from this subgroup of order 8?

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