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I've been stuck on a problem for days now and I can't find any helpful examples anywhere. So we have $A_4$, the group of symmetries of a regular tertahedron. We color two sides red and two sides white (let's say...faces 1 and 2 are red and faces 3 and 4 are white). What are the elements of the stabilizer $Z \le A_4$ of this tetrahedron?

I'm just uncertain how to think of this; is the stabilizer just {e}, since this is the only element of $A_4$ that keeps any given face in the same place and at the same time keeps the tetrahedron colored in the same way, or is it {e,(1,2)(3,4)}, since the rotation (1,2)(3,4) keeps the tetrahedron colored in the exact same way, even though the two red faces and the two white faces are swapped?

Any help will be GREATLY appreciated!

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  • $\begingroup$ The correct stabiliser is the second group you mentioned (i.e. {e, (12)(34)}). This gives $Z_2$ (or, by the pens of serious group theorists, just $2$). $\endgroup$
    – Parcly Taxel
    Dec 1, 2017 at 11:00

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The eternal problem in combinatorics: What does "the same" (and, by extension, "different") really mean?

In this case, I would think that they meant "colored the same way". It is the interpretation that makes the most sense to me in this context.

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