Explore BrainMass

Central Extension Problem

Describe all nonisomorphic central extensions of Z_n by Z_2 x Z_2 meaning central group extensions of the following form

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

Meaning, determine those nonisomorphic groups G that can be described by such an extension. Please also explain how you came up with the answer.

Solution Preview

The fact that Z_n is the kernel of the given inclusion amounts to the existence in G of an element a of order n (so that a generates Z_n). We know that G/<a> is Z_2xZ_2 so it is generated by two elements b,c of order 2 that commute. So G is any such group generated by a,b,c. Let's see what such groups we have:

Because Z_n is normal in G we know that b*a*(b^-1) belongs to Z_n (I denoted by * the group operation, and by b^-1 the inverse ...

Solution Summary

A central extension problem is solved.