- 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
submodule s of "M". It is denoted Soc("M"). In particular, a module issemisimple 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 thesubgroup generated by the minimal non-trivialnormal subgroup s of "G". The socle is a direct product of minimal normal subgroups. As an example, consider thecyclic 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
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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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