Classify Z4 Z6 according to the Fundamental Theorem of Finit

Classify (Z₄ Z₆)/<(2,2)>, according to the Fundamental Theorem of Finite Abelian groups.

Solution

This group Z4 Z6 is an alebian group of order 24.

Further, The group Z4 is not a semisimple Z-module. First, Z4 is not a simple group. Secondly, it cannot be written nontrivially as a direct sum of any subgroups, since its subgroups lie in a chain Z4 2Z4 (0), and no two proper nonzero subgroups intersect in (0)

and the group Z6 is a semisimple Z-module.

Define f : Z6 Z2 Z3 by setting f(0) = (0, 0), f(1) = (1, 1), f(2) = (0, 2), f(3) = (1, 0), f(4) = (0, 1), f(5) = (1, 2). This defines an isomorphism, showing that Z6 is isomorphic to a direct sum of simple abelian groups

[Because, A module M is said to be semisimple if it can be expressed as a sum (possibly infinite) of simple submodules]