# Nonisomorphic Central Extensions

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Describe all nonisomorphic central extensions of Z_2 x Z_2 by a cyclic group Z_n for arbitrary n, meaning central extensions of the form:

1 --> Z_n --> G --> Z_2 x Z_2 --> 1

Â© BrainMass Inc. brainmass.com December 24, 2021, 4:49 pm ad1c9bdddfhttps://brainmass.com/math/linear-algebra/nonisomorphic-central-extensions-9446

#### Solution Preview

There is a theorem stating that for an extension like this:

1-->H-->G-->M-->1

If H is an abelian group then all the extensions can be found through analyzing the second homologies in the form of H^2(M, H).

We also know that if m and n are relatively prime then Z_m x Z_n= Z_(mxn).

Well, now we want G such ...

#### Solution Summary

Nonisomorphic central extensions are described.

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