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Set-Theoretic Operations in a Group
Set-Theoretic Operations in a Group
Subsections
Functions Relating to Group Order
Exponent(G) : GrpPC -> RngIntElt
The exponent of the group G.
FactoredIndex(G, H) : GrpPC, GrpPC -> [<RngIntElt, RngIntElt>]
The factored index of the subgroup H in the group G.
FactoredOrder(G) : GrpPC -> [<RngIntElt, RngIntElt>]
The factored order of the group G.
Index(G, H) : GrpPC, GrpPC -> RngIntElt
The index of the subgroup H in the group G, returned as an
ordinary integer.
Order(G) : GrpPC -> RngIntElt
# G : GrpPC -> RngIntElt
The order of the group G, returned as an ordinary integer.
Membership and Equality
g in G : GrpPCElt, GrpPC -> BoolElt
Given an element g and a group G, return true if g is an element
of G, false otherwise.
g notin G : GrpPCElt, GrpPC -> BoolElt
Given an element g and a group G, return true if g is not an
element of G, false otherwise.
S subset G : { GrpPCElt } , GrpPC -> 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 : { GrpPCElt } , GrpPC -> 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 : GrpPC, GrpPC -> BoolElt
Given groups G and H, subgroups of some covering group, return
true if H is a subgroup of G, false otherwise.
H notsubset G : GrpPC, GrpPC -> BoolElt
Given groups G and H, subgroups of some covering group, return
true if H is not a subgroup of G, false otherwise.
G eq H : GrpPC, GrpPC -> BoolElt
Given groups G and H, subgroups of some covering group, return
true if G and H are the same group, false otherwise.
G ne H : GrpPC, GrpPC -> BoolElt
Given groups G and H, subgroups of some covering group, return
true if G and H are distinct groups, false otherwise.
Set Operations
NumberingMap(G) : GrpPC -> Map
A bijective mapping from the group G onto the set of integers {1 ... |G|}.
The actual mapping depends upon the current presentation for G.
Random(G) : GrpPC -> GrpPCElt
An element, randomly chosen, from the group G.
Representative(G) : GrpPC -> GrpPCElt
Rep(G) : GrpPC -> GrpPCElt
A representative element of G.
Example GrpPC_Set (H13E3)
Given the subgroups H and K of G,
construct the set product of the groups H and K.
> set_product := func<G, H, K | { G | x * y : x in H, y in K }>;
Given a subgroup H of the pc-group G, construct H as a set
of elements of G.
> elements := func<G, H | { G | x : x in H }>;
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