Socle (mathematics)

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.

ee also

*Radical of a module

References

* cite book
last = Alperin
first = J.L.
authorlink =
coauthors = Rowen B. Bell
title = Groups and Representations
publisher = Springer-Verlag
date = 1995
location =
pages = 136
url =
doi =
id =
isbn = 0-387-94526-1

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