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.
[edit] See also
- Cosocle
