The socle of a group 
is the subgroup generated by its minimal normal
subgroups. For example, the symmetric group 
has two nontrivial normal subgroups:
and 
. But 
contains 
, so 
is the only minimal subgroup, and the socle of 
is 
.
Socle
See also
Group, Group Block, Normal Subgroup, Primitive Group, Transitive GroupThis entry contributed by Todd Rowland
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Cite this as:
Weisstein, Eric W., with contributions by Todd Rowland. "Socle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Socle.html