Set-Theoretic Operations

Subsections

• Functions Relating to Group Order
• Membership and Equality
• Set Operations

Functions Relating to Group Order

Order(G) : GrpAb -> RngIntElt

# G : GrpAb -> RngIntElt

The order of the group G, returned as an ordinary integer. If G is an infinite group, the value zero is returned.

FactoredOrder(G) : GrpAb -> [<RngIntElt, RngIntElt>]

The factored order of the group G, returned as a sequence of prime-exponent pairs. If G is an infinite group, the empty sequence is returned.

Index(G, H) : GrpAb, GrpAb -> RngIntElt

The index of the subgroup H in the group G, returned as an ordinary integer. If H has infinite index in G, the value zero is returned.

FactoredIndex(G, H) : GrpAb, GrpAb -> [<RngIntElt, RngIntElt>]

The factored index of the subgroup H in the group G, returned as a sequence of prime-exponent pairs. If H has infinite index in G, the empty sequence is returned.

Exponent(G) : GrpAb -> RngIntElt

The exponent of the group G. If the group is infinite, the value zero is returned.

IsFinite(G) : GrpAb -> BoolElt

Return true if the group G is finite.

Membership and Equality

g in G : GrpAbElt, GrpAb -> BoolElt

Given an element g and a group G, return true if g is an element of G, false otherwise.

g notin G : GrpAbElt, GrpAb -> BoolElt

Given an element g and a group G, return true if g is not an element of G, false otherwise.

S subset G : { GrpAbElt } , GrpAb -> BoolElt

Given an group G and a set S of elements belonging to a group H, where G and H have some covering group, return true if S is a subset of G, false otherwise.

S notsubset G : { GrpAbElt } , GrpAb -> BoolElt

Given a group G and a set S of elements belonging to a group H, where G and H have some covering group, return true if S is not a subset of G, false otherwise.

H subset G : GrpAb, GrpAb -> BoolElt

Given groups G and H, subgroups of some common overgroup, return true if H is a subgroup of G, and false otherwise.

H notsubset G : GrpAb, GrpAb -> BoolElt

Given groups G and H, subgroups of some common overgroup, return true if H is not a subgroup of G, and false otherwise.

G eq H : GrpAb, GrpAb -> BoolElt

Given groups G and H, subgroups of some common overgroup, return true if G and H are identical, and false otherwise.

G ne H : GrpAb, GrpAb -> BoolElt

Given groups G and H, subgroups of some common overgroup, return true if G and H are distinct groups, and false otherwise.

Set Operations

NumberingMap(G) : GrpAb -> Map

A bijective mapping from the finite group G onto the set of integers { 1 ... |G| }. The actual mapping depends upon choice of standard generators for G.

RandomProcess(G) : GrpAb -> Process

    Slots: RngIntElt                    Default: 10

    Scramble: RngIntElt                 Default: 100

Create a process to generate randomly chosen elements from the finite group G. The process is based on the product-replacement algorithm of [CLGM+95], modified by the use of an accumulator. At all times, N elements are stored where N is the maximum of the specified value for Slots and Ngens(G) + 1. Initially, these are just the generators of G. As well, one extra group element is stored, the accumulator. Initially, this is the identity. Random elements are now produced by successive calls to Random(P), where P is the process created by this function. Each such call chooses one of the elements in the slots and adds it into the accumulator. The element in that slot is replaced by the sum of it and another randomly chosen slot. The random value returned is the new accumulator value. Setting Scramble := m causes m such sum-replacement operations to be performed before the process is returned. Note that this algorithm cannot produce well-distributed random elements of an infinite group.

Random(P) : Process -> GrpAbElt

Given a random element process P created by the function RandomProcess(G) for the finite abelian group G, return the next random element of G defined by the process.

Random(G) : GrpAb -> GrpAbElt

An element chosen at random from the finite group G.

Rep(G) : GrpAb -> GrpAbElt

A representative element of G.