The Mathieu Groups (Simple Sporadic Symmetries) Scott Harper (University of St Andrews) Tomorrow's Mathematicians Today 21st February 2015 Scott Harper The Mathieu Groups 21st February 2015 1 / 15 The Mathieu Groups (Simple Sporadic Symmetries) Scott Harper (University of St Andrews) Tomorrow's Mathematicians Today 21st February 2015 Scott Harper The Mathieu Groups 21st February 2015 2 / 15 1 2 A symmetry is a structure preserving permutation of the underlying set. A group acts faithfully on an object if it is isomorphic to a subgroup of the 4 3 symmetry group of the object. Symmetry group: D4 The stabiliser of a point in a group G is Group of rotations: the subgroup of G which fixes x. ∼ h(1 2 3 4)i = C4 Subgroup fixing 1: h(2 4)i Symmetry Scott Harper The Mathieu Groups 21st February 2015 3 / 15 A symmetry is a structure preserving permutation of the underlying set. A group acts faithfully on an object if it is isomorphic to a subgroup of the symmetry group of the object. The stabiliser of a point in a group G is Group of rotations: the subgroup of G which fixes x. ∼ h(1 2 3 4)i = C4 Subgroup fixing 1: h(2 4)i Symmetry 1 2 4 3 Symmetry group: D4 Scott Harper The Mathieu Groups 21st February 2015 3 / 15 A group acts faithfully on an object if it is isomorphic to a subgroup of the symmetry group of the object. The stabiliser of a point in a group G is Group of rotations: the subgroup of G which fixes x. ∼ h(1 2 3 4)i = C4 Subgroup fixing 1: h(2 4)i Symmetry 1 2 A symmetry is a structure preserving permutation of the underlying set. 4 3 Symmetry group: D4 Scott Harper The Mathieu Groups 21st February 2015 3 / 15 A group acts faithfully on an object if it is isomorphic to a subgroup of the symmetry group of the object. The stabiliser of a point in a group G is the subgroup of G which fixes x. Subgroup fixing 1: h(2 4)i Symmetry 1 2 A symmetry is a structure preserving permutation of the underlying set. 4 3 Symmetry group: D4 Group of rotations: ∼ h(1 2 3 4)i = C4 Scott Harper The Mathieu Groups 21st February 2015 3 / 15 The stabiliser of a point in a group G is the subgroup of G which fixes x. Subgroup fixing 1: h(2 4)i Symmetry 1 2 A symmetry is a structure preserving permutation of the underlying set. A group acts faithfully on an object if it is isomorphic to a subgroup of the 4 3 symmetry group of the object. Symmetry group: D4 Group of rotations: ∼ h(1 2 3 4)i = C4 Scott Harper The Mathieu Groups 21st February 2015 3 / 15 The stabiliser of a point in a group G is the subgroup of G which fixes x. Symmetry 1 2 A symmetry is a structure preserving permutation of the underlying set. A group acts faithfully on an object if it is isomorphic to a subgroup of the 4 3 symmetry group of the object. Symmetry group: D4 Group of rotations: ∼ h(1 2 3 4)i = C4 Subgroup fixing 1: h(2 4)i Scott Harper The Mathieu Groups 21st February 2015 3 / 15 Symmetry 1 2 A symmetry is a structure preserving permutation of the underlying set. A group acts faithfully on an object if it is isomorphic to a subgroup of the 4 3 symmetry group of the object. Symmetry group: D4 The stabiliser of a point in a group G is Group of rotations: the subgroup of G which fixes x. ∼ h(1 2 3 4)i = C4 Subgroup fixing 1: h(2 4)i Scott Harper The Mathieu Groups 21st February 2015 3 / 15 Definition A non-trivial group is called simple if it has no proper non-trivial normal subgroups. The cyclic groups of prime order are simple. The alternating groups of degree at least five are simple. Simplicity Scott Harper The Mathieu Groups 21st February 2015 4 / 15 The cyclic groups of prime order are simple. The alternating groups of degree at least five are simple. Simplicity Definition A non-trivial group is called simple if it has no proper non-trivial normal subgroups. Scott Harper The Mathieu Groups 21st February 2015 4 / 15 The alternating groups of degree at least five are simple. Simplicity Definition A non-trivial group is called simple if it has no proper non-trivial normal subgroups. The cyclic groups of prime order are simple. Scott Harper The Mathieu Groups 21st February 2015 4 / 15 Simplicity Definition A non-trivial group is called simple if it has no proper non-trivial normal subgroups. The cyclic groups of prime order are simple. The alternating groups of degree at least five are simple. Scott Harper The Mathieu Groups 21st February 2015 4 / 15 Shuffles which can be obtained by an even number of transpositions (i.e. two-card swaps). Alternating group A12. Alternating Groups: Shuffles A pack of 12 cards. Scott Harper The Mathieu Groups 21st February 2015 5 / 15 Alternating group A12. Alternating Groups: Shuffles A pack of 12 cards. Shuffles which can be obtained by an even number of transpositions (i.e. two-card swaps). Scott Harper The Mathieu Groups 21st February 2015 5 / 15 Alternating Groups: Shuffles A pack of 12 cards. Shuffles which can be obtained by an even number of transpositions (i.e. two-card swaps). Alternating group A12. Scott Harper The Mathieu Groups 21st February 2015 5 / 15 Twisting group: Alternating group A12 Alternating Groups: Kissing Spheres Scott Harper The Mathieu Groups 21st February 2015 6 / 15 Twisting group: Alternating group A12 Alternating Groups: Kissing Spheres Scott Harper The Mathieu Groups 21st February 2015 6 / 15 Alternating Groups: Kissing Spheres Twisting group: Alternating group A12 Scott Harper The Mathieu Groups 21st February 2015 6 / 15 A transposition of the contents of space p and space q, written [p; q], is called an elementary move. This is valid if one of p or q is empty. Example 1: [16; 15] cannot be followed by [12; 11]. Example 2: [16; 12] can be followed by [12; 11]. Example 3: The 3-cycle (11 12 15) = Associated group: [16; 12][12; 11][11; 15][15; 16] is valid Alternating group A15 and fixes the empty space at 16. Alternating Groups: A Puzzle Scott Harper The Mathieu Groups 21st February 2015 7 / 15 Example 1: [16; 15] cannot be followed by [12; 11]. Example 2: [16; 12] can be followed by [12; 11]. Example 3: The 3-cycle (11 12 15) = Associated group: [16; 12][12; 11][11; 15][15; 16] is valid Alternating group A15 and fixes the empty space at 16. Alternating Groups: A Puzzle A transposition of the contents of space p and space q, written [p; q], is called an elementary move. This is valid if one of p or q is empty. Scott Harper The Mathieu Groups 21st February 2015 7 / 15 Example 2: [16; 12] can be followed by [12; 11]. Example 3: The 3-cycle (11 12 15) = Associated group: [16; 12][12; 11][11; 15][15; 16] is valid Alternating group A15 and fixes the empty space at 16. Alternating Groups: A Puzzle A transposition of the contents of space p and space q, written [p; q], is called an elementary move. This is valid if one of p or q is empty. Example 1: [16; 15] cannot be followed by [12; 11]. Scott Harper The Mathieu Groups 21st February 2015 7 / 15 Example 3: The 3-cycle (11 12 15) = Associated group: [16; 12][12; 11][11; 15][15; 16] is valid Alternating group A15 and fixes the empty space at 16. Alternating Groups: A Puzzle A transposition of the contents of space p and space q, written [p; q], is called an elementary move. This is valid if one of p or q is empty. Example 1: [16; 15] cannot be followed by [12; 11]. Example 2: [16; 12] can be followed by [12; 11]. Scott Harper The Mathieu Groups 21st February 2015 7 / 15 Associated group: Alternating group A15 Alternating Groups: A Puzzle A transposition of the contents of space p and space q, written [p; q], is called an elementary move. This is valid if one of p or q is empty. Example 1: [16; 15] cannot be followed by [12; 11]. Example 2: [16; 12] can be followed by [12; 11]. Example 3: The 3-cycle (11 12 15) = [16; 12][12; 11][11; 15][15; 16] is valid and fixes the empty space at 16. Scott Harper The Mathieu Groups 21st February 2015 7 / 15 Alternating Groups: A Puzzle A transposition of the contents of space p and space q, written [p; q], is called an elementary move. This is valid if one of p or q is empty. Example 1: [16; 15] cannot be followed by [12; 11]. Example 2: [16; 12] can be followed by [12; 11]. Example 3: The 3-cycle (11 12 15) = Associated group: [16; 12][12; 11][11; 15][15; 16] is valid Alternating group A15 and fixes the empty space at 16. Scott Harper The Mathieu Groups 21st February 2015 7 / 15 transitively if for all x; y 2 X there exists g 2 G such that xg = y; k-transitively if for all sequences of distinct points k (x1;:::; xk ); (y1;:::; yk ) 2 X there exists g 2 G such that xi g = yi ; sharply k-transitively if the above g is the unique such element.