Operations on Elements

Equality and Comparison

u eq v: GrpBBElt, GrpBBElt -> BoolElt

Returns true if and only if the underlying concrete group elements for \(u\) and \(v\) are equal.

u ne v: GrpBBElt, GrpBBElt -> BoolElt

Returns true if and only if the underlying concrete group elements for \(u\) and \(v\) are not equal.

Attributes of Elements

Parent(u): GrpBBElt -> GrpBB

The parent group \(G\) of the element \(u\).

UnderlyingElement(u): GrpBBElt -> GrpElt

The concrete group element corresponding to the BB-group element \(u\).

Order(u): GrpBBElt -> RngIntElt

The order of the underlying concrete group element of \(u\).

Example: Standard Generators of M24 (ex-124120)

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The following function takes a black box group isomorphic to \(M_{24}\) and finds standard generators. It is taken from the ATLAS of Finite Group Representations page on \(M_{24}\).

> m24_standard := function(B)
> repeat a := PseudoRandom(B); until Order(a) eq 10;
> a := a ^ 5;
> repeat b := PseudoRandom(B); until Order(b) eq 15;
> b := b ^ 5;
> repeat b := b ^ PseudoRandom(B); ab := a*b;
> until Order(ab) eq 23;
> x := ab*(ab^2*b)^2*ab*b;
> if Order(x) eq 5 then b := b^-1; end if;
> return a,b;
> end function;

We take a group which must be \(M_{24}\) and find these generators.

> G := PermutationGroup<24 |
> [ 20, 4, 10, 3, 15, 9, 7, 1, 11, 22, 21, 19, 8, 2, 24, 5,
> 12, 18, 13, 16, 14, 23, 6, 17 ],
> [ 12, 18, 3, 2, 7, 11, 5, 21, 19, 22, 23, 1, 14, 17, 10,
> 8, 4, 13, 24, 20, 9, 15, 6, 16 ]>;
> #G;
244823040
> Transitivity(G);
5
> B := NaturalBlackBoxGroup(G);
> a,b := m24_standard(B); a,b;
GrpBBElt (1, 16)(2, 22)(3, 14)(4, 15)(5, 11)(6, 24)(7,
10)(8, 18)(9, 19)(12, 17)(13, 20)(21, 23)
GrpBBElt (1, 14, 17)(2, 18, 13)(5, 16, 20)(7, 22, 9)(8, 24,
15)(19, 23, 21)

The printing of the GrpBBElts shows the underlying concrete group elements. These may be extracted using the UnderlyingElement intrinsic for use within \(G\).

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