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

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!

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What do "congruence-permutable" and "congruence-distributive" mean? –  Chris Eagle Jan 25 '13 at 10:44

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, $M_3$.