How many congruence relations does a 4-element non-cyclic group have?
Am I right that I have to find the normal subgroups in order to find the congruence relations?
Thanks
2026-03-26 07:59:27.1774511967
Number of congruence relations of a 4-element non-cyclic group
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The 4-element non-cyclic group is Klein four-group. An example of this group is the group $\mathcal{P}(S)$ where $S = \{a,b\}$ with the symmetric difference as the binary operation.
There are five congruences on this group by counting the normal subgroups which two of them are trivial normal subgroups. The three non-trivial ones are $\{\varnothing,\{a\}\}$, $\{\varnothing,\{b\}\}$ and $\{\varnothing,\{a,b\}\}$