Socle
The socle of a group G is the subgroup generated by its minimal normal subgroups. For example, the symmetric group S_4 has two nontrivial normal subgroups: A_4 and N={{1,2,3,4},{2,1,4,3},{3,4,1,2},{4,3,2,1}}. But A_4 contains N, so N is the only minimal subgroup, and the socle of S_4 is N.
See also
Group, Group Block, Normal Subgroup, Primitive Group, Transitive GroupThis entry contributed by Todd Rowland
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Rowland, Todd. "Socle." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Socle.html