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Set-Theoretic Operations

Set-Theoretic Operations

Subsections

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
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 abelian group G. The process uses an `expansion' procedure to construct a set of elements corresponding to fairly long words in the generators of G. At all times, N elements are stored where N is the maximum of Slots and Ngens(G) + 1. Initially, these are just the generators of G and products of pairs of generators of G. Random elements are now produced by successive calls to Random(P), where P is the process created by this function. Each such call chooses an element from the process and returns it, replacing it with the product of it and another random element (on the left or the right). Before the process is returned, Scramble such operations are performed.
Random(P) : Process -> GrpAbElt
Given a random element process P created by the function RandomProcess(G) for the finite abelian group G, construct a random element of G by forming a random product over the expanded generating set stored as part of the process. The expanded generating set stored with the process is modified by replacing an existing generator by the element returned.
Random(G) : GrpAb -> GrpAbElt
An element chosen at random from the finite group G.
Rep(G) : GrpAb -> GrpAbElt
A representative element of G.
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