Group#

class libcasm.group.Group(
elements: list[Any] | None = None,
multiplication_table: list[list[int]] | None = None,
head_group: Group | None = None,
head_group_index: set[int] | None = None,
)[source]#

Bases: GenericGroup

A group with user specified elements.

Constructor

Notes

To construct a head group, provide elements and multiplication_table. To construct a subgroup, provide head_group and head_group_index and optionally, elements.

Examples

Construct a head group with 4 elements and the following multiplication table:

elements = ["e", "a", "b", "c"]

multiplication_table = [
    [0, 1, 2, 3],  # e * e = e, e * a = a, e * b = b, e * c = c
    [1, 0, 3, 2],  # a * e = a, a * a = e, a * b = c, a * c = b
    [2, 3, 0, 1],  # b * e = b, b * a = c, b * b = e, b * c = a
    [3, 2, 1, 0],  # c * e = c, c * a = b, c * b = a, c * c = e
]

group = Group(
    elements=elements,
    multiplication_table=multiplication_table,
)

Construct a subgroup of the above group with head group indices {0, 2}, corresponding to the elements [“e”, “b”].

subgroup = group.make_subgroup(
    head_group_indices={0, 2}
)

If the original group represents a factor group, then the subgroup can be represented as a subgroup of the factor group with the same head group indices, but with different elements, for instance [“e”, “b’”] where b’ is a different member of the same coset as b in the original group.

subgroup = group.make_subgroup(
    head_group_indices={0, 2},
    elements=["e", "b'"],
)
Parameters:

  • elements (Optional[list[Any]] = None) – The group elements. The type of the elements is not specified, but they should be compatible with the multiplication table. The first element must be the identity element.

  • multiplication_table (Optional[list[list[int]]] = None) –

    The multiplication table element multiplication_table[i][j] == k represents that np.allclose(elements[k], elements[i] @ elements[j]) == True.

    The multiplication table must be square, represent a closed group, and be consistent with having identity operation as the first element.

  • head_group (Optional[GenericGroup] = None) – The head group if this is a subgroup, else None. If None, then this is a head group.

  • head_group_index (Optional[set[int]] = None) – Indices of elements in the head group (which may or may not be self) to include in a subgroup. Only used if head_group is not None.

Methods

class_of(…)

Returns the index of the conjugacy class containing an element

conjugacy_classes(…)

Returns the conjugacy classes

from_dict(…)

Construct a GenericGroup from a Python dict.

inv(…)

Returns the index of the inverse of an element

make_subgroup(…)

Make a subgroup of this group with the specified head group indices.

mult(…)

Returns the index of the element product.

size(…)

Returns the size of the group

to_dict(…)

Represent the GenericGroup as a Python dict

Attributes

head_group

The head group if this is a subgroup, else None.

head_group_index

The list of head group indices (guaranteed sorted)

inverse_index

The list of inverse indices

is_subgroup

True if this is a subgroup, False otherwise.

multiplication_table

The multiplication table.

elements

The group elements.
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