Permutation group. Show that the group has just three members of length 2 in cycle notation.

Question

Show that there are just three members of $S_4$ which have two cycles of length 2 when written in cycle notation.

I'm having trouble figuring out how I should test this. It seems dumb to write out all 24 permutation combinations and then convert them to cycle notation.

score 0 · Accepted Answer · answered Apr 03 '18 at 18:49

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Assuming (and we can. Why?) the cycles are pairwise disjoint, it is clear the only such elements are $\;(12)(34),\,(13)(24),\,(14)(23)\;$ .

answered Apr 03 '18 at 18:49

DonAntonio

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Assuming the group contains all inverses, why don't we count for example (21)(43)? – Elias Apr 04 '18 at 14:33
@Elias Because we always have $;(ik)=(ki);,;;i\neq k;$ ... – DonAntonio Apr 04 '18 at 17:04

Permutation group. Show that the group has just three members of length 2 in cycle notation.

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