PermutationGroupExamples

permgrps.spad line 1007 [edit on github]

PermutationGroupExamples provides permutation groups for some classes of groups: symmetric, alternating, dihedral, cyclic, direct products of cyclic, which are in fact the finite abelian groups of symmetric groups called Young subgroups. Furthermore, Rubik's group as permutation group of 48 integers and a list of sporadic simple groups derived from the atlas of finite groups.

abelianGroup : List(PositiveInteger) -> PermutationGroup(Integer)

abelianGroup([n1, ..., nk]) constructs the abelian group that is the direct product of cyclic groups with order ni.

alternatingGroup : List(Integer) -> PermutationGroup(Integer)

alternatingGroup(li) constructs the alternating group acting on the integers in the list li, generators are in general the n-2-cycle (li.3, ..., li.n) and the 3-cycle (li.1, li.2, li.3), if n is odd and product of the 2-cycle (li.1, li.2) with n-2-cycle (li.3, ..., li.n) and the 3-cycle (li.1, li.2, li.3), if n is even. Note: duplicates in the list will be removed.

alternatingGroup : PositiveInteger -> PermutationGroup(Integer)

alternatingGroup(n) constructs the alternating group An acting on the integers 1, ..., n, generators are in general the n-2-cycle (3, ..., n) and the 3-cycle (1, 2, 3) if n is odd and the product of the 2-cycle (1, 2) with n-2-cycle (3, ..., n) and the 3-cycle (1, 2, 3) if n is even.

cyclicGroup : List(Integer) -> PermutationGroup(Integer)

cyclicGroup([i1, ..., ik]) constructs the cyclic group of order k acting on the integers i1, ..., ik. Note: duplicates in the list will be removed.

cyclicGroup : PositiveInteger -> PermutationGroup(Integer)

cyclicGroup(n) constructs the cyclic group of order n acting on the integers 1, ..., n.

dihedralGroup : List(Integer) -> PermutationGroup(Integer)

dihedralGroup([i1, ..., ik]) constructs the dihedral group of order 2k acting on the integers out of i1, ..., ik. Note: duplicates in the list will be removed.

dihedralGroup : PositiveInteger -> PermutationGroup(Integer)

dihedralGroup(n) constructs the dihedral group of order 2n acting on integers 1, ..., N.

janko2 : () -> PermutationGroup(Integer)

janko2 constructs the janko group acting on the integers 1, ..., 100.

janko2 : List(Integer) -> PermutationGroup(Integer)

janko2(li) constructs the janko group acting on the 100 integers given in the list li. Note: duplicates in the list will be removed. Error: if li has less or more than 100 different entries

mathieu11 : () -> PermutationGroup(Integer)

mathieu11 constructs the mathieu group acting on the integers 1, ..., 11.

mathieu11 : List(Integer) -> PermutationGroup(Integer)

mathieu11(li) constructs the mathieu group acting on the 11 integers given in the list li. Note: duplicates in the list will be removed. error, if li has less or more than 11 different entries.

mathieu12 : () -> PermutationGroup(Integer)

mathieu12 constructs the mathieu group acting on the integers 1, ..., 12.

mathieu12 : List(Integer) -> PermutationGroup(Integer)

mathieu12(li) constructs the mathieu group acting on the 12 integers given in the list li. Note: duplicates in the list will be removed Error: if li has less or more than 12 different entries.

mathieu22 : () -> PermutationGroup(Integer)

mathieu22 constructs the mathieu group acting on the integers 1, ..., 22.

mathieu22 : List(Integer) -> PermutationGroup(Integer)

mathieu22(li) constructs the mathieu group acting on the 22 integers given in the list li. Note: duplicates in the list will be removed. Error: if li has less or more than 22 different entries.

mathieu23 : () -> PermutationGroup(Integer)

mathieu23 constructs the mathieu group acting on the integers 1, ..., 23.

mathieu23 : List(Integer) -> PermutationGroup(Integer)

mathieu23(li) constructs the mathieu group acting on the 23 integers given in the list li. Note: duplicates in the list will be removed. Error: if li has less or more than 23 different entries.

mathieu24 : () -> PermutationGroup(Integer)

mathieu24 constructs the mathieu group acting on the integers 1, ..., 24.

mathieu24 : List(Integer) -> PermutationGroup(Integer)

mathieu24(li) constructs the mathieu group acting on the 24 integers given in the list li. Note: duplicates in the list will be removed. Error: if li has less or more than 24 different entries.

rubiksGroup : () -> PermutationGroup(Integer)

rubiksGroup constructs the permutation group representing Rubic's Cube acting on integers 10*i+j for 1 <= i <= 6, 1 <= j <= 8. The faces of Rubik's Cube are labelled in the obvious way Front, Right, Up, Down, Left, Back and numbered from 1 to 6 in this given ordering, the pieces on each face (except the unmoveable center piece) are clockwise numbered from 1 to 8 starting with the piece in the upper left corner. The moves of the cube are represented as permutations on these pieces, represented as a two digit integer ij where i is the numer of theface (1 to 6) and j is the number of the piece on this face. The remaining ambiguities are resolved by looking at the 6 generators, which represent a 90 degree turns of the faces, or from the following pictorial description. Permutation group representing Rubic's Cube acting on integers 10*i+j for 1 <= i <= 6, 1 <= j <=8. Rubik's Cube: +-----+ +-- B where: marks Side # : / U /|/ / / | F(ront) <-> 1 L --> +-----+ R| R(ight) <-> 2 | | + U(p) <-> 3 | F | / D(own) <-> 4 | |/ L(eft) <-> 5 +-----+ B(ack) <-> 6 ^ | DThe Cube's surface: The pieces on each side +---+ (except the unmoveable center |567| piece) are clockwise numbered |4U8| from 1 to 8 starting with the |321| piece in the upper left +---+---+---+ corner (see figure on the |781|123|345| left). The moves of the cube |6L2|8F4|2R6| are represented as |543|765|187| permutations on these pieces. +---+---+---+ Each of the pieces is |123| represented as a two digit |8D4| integer ij where i is the |765| # of the side ( 1 to 6 for +---+ F to B (see table above )) |567| and j is the # of the piece. |4B8| |321| +---+

symmetricGroup : List(Integer) -> PermutationGroup(Integer)

symmetricGroup(li) constructs the symmetric group acting on the integers in the list li, generators are the cycle given by li and the 2-cycle (li.1, li.2). Note: duplicates in the list will be removed.

symmetricGroup : PositiveInteger -> PermutationGroup(Integer)

symmetricGroup(n) constructs the symmetric group Sn acting on the integers 1, ..., n, generators are the n-cycle (1, ..., n) and the 2-cycle (1, 2).

youngGroup : List(Integer) -> PermutationGroup(Integer)

youngGroup([n1, ..., nk]) constructs the direct product of the symmetric groups Sn1, ..., Snk.

youngGroup : Partition -> PermutationGroup(Integer)

youngGroup(lambda) constructs the direct product of the symmetric groups given by the parts of the partition lambda.