MAGMA Computational Algebra System

Properties of Groups of Lie Type

IsFinite(G) : GrpLie -> BoolElt

Return true if and only if the group of Lie type G is finite.

IsAbelian(G) : GrpLie -> BoolElt

Returns true if the group of Lie type G is abelian.

IsSimple(G) : GrpLie -> BoolElt

Returns true if the group of Lie type G is a simple group as an algebraic group, ie, G has no proper connected normal subgroups. This is true if, and only if, the underlying root datum is irreducible. Note that this does not usually mean that G is simple as an abstract group. In previous releases of Magma this function was incorrectly called IsIrreducible.

IsSimplyLaced(G) : GrpLie-> BoolElt

Returns true if the group of Lie type G is simply laced, i.e. its Dynkin diagram contains no multiple bonds.

IsSemisimple(G) : GrpLie-> BoolElt

Returns true if the group of Lie type G is semisimple.

IsAdjoint(G) : GrpLie-> BoolElt

Returns true if the group of Lie type G is adjoint(ie. the isogeny group is trivial).

IsSimplyConnected(G) : GrpLie-> BoolElt

Returns true if the group of Lie type G is simply connected(ie. the isogeny group is equal to the fundamental group, ie. the coisogeny group is trivial).

IsSplit(G) : GrpLie -> BoolElt

Returns true if and only if the group of Lie type G is split.

IsTwisted(G) : GrpLie -> BoolElt

Returns true if and only if the group of Lie type G is twisted.