Subsections

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

The exponent of the group G.

The factored index of the subgroup H in the group G.

The factored order of the group G.

The index of the subgroup H in the group G, returned as an ordinary integer.

# G : GrpPC -> RngIntElt

The order of the group G, returned as an ordinary integer.

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

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

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.

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.

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

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

Given groups G and H, subgroups of some covering group, return true if G and H are the same group, false otherwise.

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

A bijective mapping from the group G onto the set of integers {1 ... |G|}. The actual mapping depends upon the current presentation for G.

Rep(G) : GrpPC -> GrpPCElt

A representative element of G.

    Slots: RngIntElt                    Default: 10

    Scramble: RngIntElt                 Default: 20

Create a process to generate randomly chosen elements from the 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 the specified value for 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). Setting Scramble := m causes m such operations to be performed before the process is returned.

Given a random element process P created by the function RandomProcess(G) for the finite group G, construct a random element of G by forming a random product over the expanded generating set constructed when the process was created.

Random(G) : GrpPC -> GrpPCElt

An element, randomly chosen, from the group G.

> set_product := func<G, H, K | { G | x * y : x in H, y in K }>;

> elements := func<G, H | { G | x : x in H }>;