(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [itex]H[/itex] be all the elements of the alternating group [itex]A_4[/itex] of exponent 2. Show that [itex]A_4[/itex] is a split extension of [itex]H[/itex] by [itex]\mathbb{Z}_3[/itex].

2. Relevant equations

None.

3. The attempt at a solution

I have already shown that [itex]H[/itex] is normal in [itex]A_4[/itex]. I also have the proof of the following proposition:

Let [itex]H[/itex] and [itex]K[/itex] be groups whose orders are relatively prime and let [itex]f:G \rightarrow K[/itex] be an extension of [itex]H[/itex] by [itex]K[/itex]. Then the extension splits iff [itex]G[/itex] has a subgroup of order [itex]|K|[/itex].

Let [itex]H=H[/itex] (as defined in the problem statement), let [itex]K=\mathbb{Z}_3[/itex], and let [itex]G=A_4[/itex]. Then certainly the orders of [itex]H[/itex] and [itex]K[/itex] are relatively prime. Also, [itex]A_4[/itex] certainly has a subgroup of order 3. Just take the identity, a product of disjoint transpositions, and its inverse and you have a subgroup. So referring back to the definition of an extension, I've reduced the problem to finding a surjective homomorphism [itex]f: A_4 \rightarrow \mathbb{Z}_3[/itex] with [itex]\ker(f)=H[/itex].

Now I'm stuck. I know I need the homomorphism to map all 4 elements of [itex]H[/itex] to 0 in [itex]\mathbb{Z}_3[/itex]. But there doesn't seem to be enough variety in the remaining elements of [itex]A_4[/itex] for me to decide which of them should be mapped to 1 and which should be mapped to 2.

Little help?

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# Homework Help: The Group A_4 as a Split Extension.

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