Every group and ring is congruence-permutable , but not necessarily congruence-distributive

Question

The problem is:

Show that every group and ring is congruence-permutable, but not necessarily congruence-distributive.

I know that in group every normal sub group has permutable property and in every ring, every ideal has permutable property. Moreover, for not being distributive I should give an example. please Guide me.

Thanks!

Answer 1

If you compute the entire lattice of normal subgroups of the Klein 4-group, $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$, you will see you get the smallest modular non-distributive lattice, $M_3$.