Socle (mathematics)

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In mathematics, the term socle has two distinct but related meanings.

In the context of a module M over a ring R, the socle of M is the sum of the minimal non-trivial 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. The socle consists precisely of the elements annihilated by the radical of R.

In the context of group theory, the socle of a group G, denoted Soc(G), is the subgroup generated by the minimal non-trivial normal subgroups of G. The socle is a direct product of minimal normal subgroups. As an example, consider the cyclic group Z₁₂ with generator u, which has two minimal normal subgroups, one generated by u ⁴ and the other by u ⁶. Thus the socle of Z₁₂ is the group generated by u ⁴ and u ⁶, which is just the group generated by u ².

[edit] See also

• Injective hull
• Radical of a module

[edit] References

This article needs additional citations for verification. Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (June 2008)

• Alperin, J.L.; Rowen B. Bell (1995). Groups and Representations. Springer-Verlag. p. 136. ISBN 0-387-94526-1.