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The Mathieu Groups (Simple Sporadic Symmetries) 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.
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