construct the socle of a group
construct the cosocle of a group
Socle( G )
Cosocle( G )
a permutation group
The socle of a group G is the subgroup generated by the minimal normal (non-trivial) subgroups of G.
The cosocle of a group G is the intersection of the maximal normal subgroups of G. It is also equal to the set of "normal non-generators" of G, that is, the set of elements of G that can be omitted from any set X for which G is the normal closure of X.
The Socle( G ) command constructs the socle of a group G.
The Cosocle( G ) command constructs the cosocle of the group G.
S ≔ Socle⁡Symm⁡4
df ≔ DirectFactors⁡S
S ≔ Socle⁡Alt⁡6
G ≔ DirectProduct⁡Alt⁡5,Alt⁡5
G≔ < a permutation group on 10 letters with 4 generators >
The cosocle of a cyclic group is trivial if, and only if, the group has square-free order.
The GroupTheory[Socle] and GroupTheory[Cosocle] commands were introduced in Maple 2019.
For more information on Maple 2019 changes, see Updates in Maple 2019.
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