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, and only if, the group of Lie type \(G\) is adjoint (i.e. the isogeny group is trivial).

IsWeaklyAdjoint(G): GrpLie -> BoolElt

Returns true if, and only if, the group of Lie type \(G\) is weakly adjoint, i.e. its isogeny group is isomorphic to \({\mathbb{Z}}^n\), where \(n\) is the difference between the rank and the semisimple rank of \(G\). Note that if \(G\) is semisimple then this function is identical to IsAdjoint.

IsSimplyConnected(G): GrpLie -> BoolElt

Returns true if, and only if, the group of Lie type \(G\) is simply connected (i.e. the isogeny group is equal to the fundamental group, i.e. the coisogeny group is trivial).

IsWeaklySimplyConnected(G): GrpLie -> BoolElt

Returns true if, and only if, the group of Lie type \(G\) is weakly simply connected, i.e. its coisogeny group is isomorphic to \({\mathbb{Z}}^n\), where \(n\) is the difference between the rank and the semisimple rank of \(G\). Note that if \(G\) is semisimple then this function is identical to IsSimplyConnected.

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.