Using the Orbit-Stabiliser Theorem or otherwise, prove that $S_4= \langle(1,2), (1,2,3,4) \rangle$? I am having a little bit of difficulty with this question, because I am not sure what it is asking. Should I consider working out the order of the stabiliser and orbits explicitly? Kindest regards.
Application of the Orbit-Stabiliser Theorem
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group-theory
finite-groups
permutations
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2Your group (allegedly it's $S_4$, but your job is to verify this) acts on $\{1, 2, 3, 4\}$ in the way you'd expect, so it's a subgroup of $S_4$. If you can find something with an orbit of size $4$ and stabilizer subgroup of order $6$, then since $S_4$ has a unique subgroup of order $24$... – 2017-02-18
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0@pjs36 isn't it the case that the stabiliser of G is simply the identity therefore orbit must have a size 24? – 2017-02-19
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0@CatherineDrysdale Orbits are subsets of $X$, the set of elements we are permuting. Here $X = \{ 1, 2, 3, 4 \}$, hence $|X| = 4$ and the largest possible length of an orbit must then be $4$, not $24$. – 2017-02-20