# Nontrivial Central Extension

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Find a nontrivial central Z_2 extension of the group A_4, meaning an extension of the form:

1 --> Z_2 --> G --> A_4 --> 1

Also, is it unique?

The trivial extension is just the direct product of Z_2 and A_4.

Â© BrainMass Inc. brainmass.com December 24, 2021, 4:49 pm ad1c9bdddfhttps://brainmass.com/math/basic-algebra/nontrivial-central-extension-9444

#### Solution Preview

First we consider the fact that A_4 has a normal series Z_2 x Z_2/Z_3 and actually we can rewrite it as 1-->Z_2 x Z_2-->A_4-->Z_3-->1. We define the operator f: A_4-->Z_3 to be a homomorphism and then consider:

G={(a,b)in A_4xZ_6; f(a)=b'}

Where b'in Z_3 is the equivalent of b in Z_6 mapped to Z_3 (converting from mod 6 to mod 3). Now, we can see that G can be defined to have the same properties provided we consider the map G-->A_4 as h(a,b)=a and then the map Z_2-->G as k(0)=(e,0) and ...

#### Solution Summary

A nontrivial central extension is dealt with.

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