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