Socle (mathematics)

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In mathematics, given a ring R and an R-module M, the socle of M is the sum of the minimal submodules of M. It is denoted Soc(M). In particular, a module is semisimple if and only if Soc(M) = M. So the socle of a module could also be defined as the unique maximal semi-simple submodule.

In group theory, the socle of a group G, denoted Soc(G), is the subgroup generated by the minimal normal subgroups of G. Consider the cyclic group Z₁₂ with generator u, which has two minimal normal subgroups, one generated by u⁴ and the other by u⁶. Thus the socle of Z₁₂ is the group generated by u⁴ and u⁶, which correspond to the cyclic subgroup generated by u².